[PDF] - Slide 4 / 252 Throughout this unit, the Standards for Mathematical PDF Document

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Geometry

Similar Triangles & Trigonometry

2015-10-22 www.njctl.org

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· Problem Solving with Similar Triangles · Trigonometric Ratios · Special Right Triangles · Review of the Pythagorean Theorem

click on the topic to go to that section

· Similar Triangles and Trigonometry · PARCC Sample Questions · Inverse Trigonometric Ratios · Converse of the Pythagorean Theorem

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SLIDE 2

Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

thers.

MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

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Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

thers.

MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for & express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab.

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Math Practice

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Three basic approaches to real world problem solving include: · Similar Triangles · Trigonometry · Pythagorean Theorem

Problem Solving with Similar Triangles and Right Triangles Slide 5 / 252

SLIDE 3

Problem Solving with Similar Triangles

Return to the Table of Contents

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One of the oldest math problems was solved using similar right triangles. About 2600 years ago, Thales of Miletus, perhaps the first Greek mathematician, was visiting Egypt and wondered what the height was of one of the Great Pyramid of Giza. Due to the shape of the pyramid, he couldn't directly measure its height.

Shadows and Similar Triangles Slide 7 / 252

One of the oldest math problems was solved using similar right triangles. About 2600 years ago, Thales of Miletus, perhaps the first Greek mathematician, was visiting Egypt and wondered what the height was of one of the Great Pyramid of Giza. Due to the shape of the pyramid, he couldn't directly measure its height.

Shadows and Similar Triangles

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Math Practice

This example (this slides and the next 10) address MP1, MP2, MP3, MP4, MP5 & MP7

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SLIDE 4

http://www.metrolic.com/travel-guides-the-great-pyramid-of-giza-147358/

When Thales visited the Great Pyramid of Giza 2600 years ago, it was already 2000 years old. He wanted to know its height.

Shadows and Similar Triangles Slide 8 / 252

He noticed that the pyramid cast a shadow, which could be measured on the ground using a measuring rod. And he realized that the measuring rod standing vertically also cast a shadow. Based on those two observations, can you think of a way he could measure the height of the pyramid? Discuss this at your table for a minute or two.

Shadows and Similar Triangles Slide 9 / 252

What 2 facts can you recall from our study of similar triangles? Fill in the blanks below. Their angles are all conguent. Their corresponding sides are in proportion to one another.

Shadows and Similar Triangles

click click

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SLIDE 5

Draw a sketch of the pyramid being measured and its shadow....and the measuring rod and its shadow. Represent the pyramid and rod as vertical lines, with the rod being much shorter than the pyramid. You won't be able to draw them to scale, since the rod is so small compared to the pyramid, but that won't affect our thinking.

Shadows and Similar Right Triangles Slide 11 / 252 Shadows and Similar Right Triangles Slide 12 / 252 Shadows and Similar Right Triangles Slide 13 / 252

SLIDE 6

x

y x y Taking away the objects and just leaving the triangles created by the height of the object, the sunlight and the shadow on the ground, we can see these are similar triangles. All the angles are equal, so the sides must be in proportion.

Shadows and Similar Right Triangles Slide 17 / 252

2 m 1 m 120 m h height of pyramid height of rod pyramid's shadow rod's shadow =

Shadows and Similar Right Triangles

height of rod rod's shadow x pyramid's shadow height of pyramid = 1 m 2 m (120 m) h = h = 60 m

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This approach can be used to measure the height of a lot of

bjects which cast a shadow.

And, a convenient measuring device is then your height, and the length of the shadow you cast. Try doing this on the next sunny day you can get outside. Measure the height of any object which is casting a shadow by comparing the length of its shadow to the length of your

wn.

Shadows and Similar Right Triangles

Lab - Indirect Measurement Reminder - Mirrors also create indirect measurement if you are doing this lab on a cloudy day.

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SLIDE 8

This approach can be used to measure the height of a lot of

bjects which cast a shadow.

And, a convenient measuring device is then your height, and the length of the shadow you cast. Try doing this on the next sunny day you can get outside. Measure the height of any object which is casting a shadow by comparing the length of its shadow to the length of your

wn.

Shadows and Similar Right Triangles

Lab - Indirect Measurement Reminder - Mirrors also create indirect measurement if you are doing this lab on a cloudy day.

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Math Practice

The lab addresses MP1, MP2, MP4, MP5 & MP7

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1 A lamppost casts a 9 ft shadow at the same time a person 6 ft tall casts a 4 ft shadow. Find the height of the lamppost. A 6 ft B 2.7 ft C 13.5 ft D 15 ft

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1 A lamppost casts a 9 ft shadow at the same time a person 6 ft tall casts a 4 ft shadow. Find the height of the lamppost. A 6 ft B 2.7 ft C 13.5 ft D 15 ft

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Answer

C Slide 20 (Answer) / 252

SLIDE 9

2 You're 6 feet tall and you notice that your shadow at one time is 3 feet long. The shadow

f a nearby building at that same moment is 20

feet long How tall is the building?

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2 You're 6 feet tall and you notice that your shadow at one time is 3 feet long. The shadow

f a nearby building at that same moment is 20

feet long How tall is the building?

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Answer

40 feet

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3 You're 1.5 m tall and you notice that your shadow at one time is 4.8 m long. The shadow

f a nearby tree at that same moment is 35 m

long How tall is the tree?

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SLIDE 10

3 You're 1.5 m tall and you notice that your shadow at one time is 4.8 m long. The shadow

f a nearby tree at that same moment is 35 m

long How tall is the tree?

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Answer

11 m

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4 Two buildings are side by side. The 35 m tall building casts a 21 m shadow. How long will the shadow of the 8 m tall building be at the same time?

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4 Two buildings are side by side. The 35 m tall building casts a 21 m shadow. How long will the shadow of the 8 m tall building be at the same time?

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Answer

4.8 m

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SLIDE 11

Similar Triangle Measuring Device

4 cm 5 inches 3 inches 2 cm 0.5 cm

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SLIDE 12

By looking along the meter stick, you can then move the card so that a distant object fills either the 0.5 cm, 2 cm or 4 cm slot. You can then measure how far the card is from your eye, along the meter stick. This creates a similar triangle that allows you to find how far away an object of known size is, or the size of an object of known distance away.

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The altitude and base of the small isosceles triangle can be directly measured, which means that the ratio of those on the larger triangle is know. Given the size or the distance to the

bject, the other can be determined.

Similar Triangle Measuring Device

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Math Practice

This example (this slide and the next) addresses MP1, MP2, MP4, MP5 & MP7

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2 m 2 cm 324 m d 20 cm distance to tower distance to card height of tower width of slot =

Shadows and Similar Right Triangles

distance to card width of slot x height of tower distance to tower = 20 cm 2 cm (324 m) d = d = 3240 m

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SLIDE 14

5 You move to another location and the Eiffel Tower (324 m tall) now fills the 4 cm slot when the card is 48 cm from your eye. How far are you from the Eiffel Tower now?

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5 You move to another location and the Eiffel Tower (324 m tall) now fills the 4 cm slot when the card is 48 cm from your eye. How far are you from the Eiffel Tower now?

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Answer

3888 m

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6 The tallest building in the world, the Burj Kalifah in Dubai, is 830 m tall. You turn your device so that it fills the 4 cm slot when it is 29.4 cm from your eye. How far are you from the building?

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SLIDE 15

6 The tallest building in the world, the Burj Kalifah in Dubai, is 830 m tall. You turn your device so that it fills the 4 cm slot when it is 29.4 cm from your eye. How far are you from the building?

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Answer

8200 m

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7 The width of a storage tank fills the 2 cm slot when the card is 48 cm from your eye. You know that the tank is 680 m away. What is its width?

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7 The width of a storage tank fills the 2 cm slot when the card is 48 cm from your eye. You know that the tank is 680 m away. What is its width?

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Answer

28 m

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SLIDE 16

8 The moon has a diameter of 3480 km. You measure it

ne night to about fill the 0.5 cm slot when the card is

54 cm from your eye. What is the distance to the moon?

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8 The moon has a diameter of 3480 km. You measure it

ne night to about fill the 0.5 cm slot when the card is

54 cm from your eye. What is the distance to the moon?

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Answer

375,840 km

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Similar Triangles and Trigonometry

1 cosθ sinθ θ Return to the Table of Contents

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SLIDE 17

Problem Solving

Recall that Thales found the height of the pyramid by using similar triangles created by the shadow of the pyramid and a rod of known length.

Slide 37 / 252 Problem Solving

But what if he were trying to solve this problem and there wasn't a shadow to use. Or you are trying to solve other types of problems which don't allow you to set up a similar triangle so easily. Trigonometry provides the needed similar triangle for any circumstance, which is why it is a powerful tool.

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So, if Thales used trig to solve his problem, he'd have considered this right triangle. First he'd measure theta, the angle between the ground and the top of the pyramid, when at a certain distance away on the ground. Then he'd imagine a similar triangle with the same angle. θ height distance

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SLIDE 18

Problem Solving with Trigonometry

He has a ready-made right triangle, thanks to mathematicians who calculated all the possible right triangles that could be created with a hypotenuse of 1 and put their measurements in a table, a trigonometry table. The side opposite the angle is named sine θ, or sinθ for short, and the side adjacent to the angle is called cosine θ,

r cosθ for short.

cosθ sinθ 1 θ θ height distance

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We know all the angles are equal since both triangles include a right angle and the angle theta, so those two angles are the same in both. And, since all the angles of a triangle total to 180 º, all three angles must be equal. Since all the angles are equal, these triangles are similar. θ height distance cosθ sinθ 1 θ

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Since all the angles are equal, the sides are in proportion, so what would this ratio be equal to in the triangle to the right?

θ height distance height distance sin θ cos θ = cos θ sin θ 1 θ

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SLIDE 19

Problem Solving with Trigonometry

Since all the angles are equal, the sides are in proportion, so what would this ratio be equal to in the triangle to the right?

θ height distance height distance sin θ cos θ = cos θ sin θ 1 θ

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Math Practice

The question on this slide addresses MP2.

Slide 42 (Answer) / 252 Problem Solving with Trigonometry

When we did the problem earlier we used the rod's height of 1 m and it's shadow's length of 2 m. That would mean that the angle between the rays of sunlight and the ground would have been 26.6º. And the length of the pyramid's shadow was 120 m. Let's use that angle and distance and see if we get the same answer. cosθ sinθ 1 26.6º 26.6º height 120 m

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If the distance was 120 m, and the angle was 26.6º, you find the height by solving for it and then using your calculator to look up the values for sin and cos. 26.6º cosθ sinθ 1 26.6º height 120 m height distance sinθ cosθ = sin(26.6º) cos(26.6º) = height (120 m) (0.448) (0.894) = (120 m) = 60 m height 120 m = sin(26.6º) cos(26.6º)

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SLIDE 20

Tangent θ

This ratio of sine to cosine is used very often, and has its own name: Tangent θ, or tanθ for short. Tangent θ is defined as Sine θ divided by Cosine θ. sinθ cosθ = tanθ height distance sinθ cosθ = Early in the last problem we found that:

Slide 45 / 252 Using Calculators with Trigonometry

The last step of that problem required finding the values of the sine and the cosine of 26.6º. When working with trigonometry, you'll need to find the values

f sine, cosine and other trig functions when given an angle.

This used to involve using tables, but now it's pretty simple to use a basic scientific calculator.

Slide 46 / 252 Using Calculators with Trigonometry

Basic scientific calculators are available on computers, tablets and smart phones. They can also be a separate device, similar to the inexpensive calculator shown

here. It can do everything you'll

need for this course.

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SLIDE 21

Using Calculators with Trigonometry

The trig functions we're going to be using right now are sine, cosine and tangent. Those are marked in the box on the calculator. On most calculators, they are noted by buttons which say SIN COS TAN

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This is for finding the sine

f an angle.

Using Calculators with Trigonometry Slide 49 / 252

This is for finding the cosine of an angle.

Using Calculators with Trigonometry Slide 50 / 252

SLIDE 22

This is for finding the tangent of an angle.

Using Calculators with Trigonometry Slide 51 / 252 Problem Solving with Trigonometry

1 sinθ cosθ θ

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In practice, we often have to measure angles of elevation

r depression in order to solve problems.

There are very accurate ways of doing that which are used by surveyors, navigators and others. But you can make a simple device, called an inclinometer, to accomplish the same thing, and then solve problems on your own.

Inclinometer Slide 53 / 252

SLIDE 23

Just tape a protractor to a meter stick and hang a small weight from the hole in the protractor. Set it up so that when the meter stick is horizontal, the string goes straight down.

Inclinometer Slide 54 / 252

Then, if you look along the meter stick, you can hold the string where it touches the protractor and read the angle. You'll have to subtract 90 degrees to get the angle to the horizon, or angle of elevation.

Inclinometer Slide 55 / 252

You are standing on the ground and look along your inclinometer to see the top of a building to be at an angle

f 30º. You then measure the distance to the base of the

building to be 30 m. Find the height of the building, remembering to add in the height your eye is above the ground.

Inclinometer Slide 56 / 252

SLIDE 24

You are standing on the ground and look along your inclinometer to see the top of a building to be at an angle

f 30º. You then measure the distance to the base of the

building to be 30 m. Find the height of the building, remembering to add in the height your eye is above the ground.

Inclinometer

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Answer height distance = tanθ height = tanθ x distance h = tan(30º)(30m) h = (0.577)(30m) h = 17.32 m + 1.52m h = 18.84 m Note: if your eyesight is 5 ft above the ground, that is 1.52 m Feet to meters = 5/3.28 = 1.52

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60 º height 200 m

You are standing 200 m away from the base of a building. You measure the top of the building to be at an angle of elevation (the angle between the ground and a line drawn to the top) of 60º. What is the height of the building?

Example Slide 57 / 252

60 º height 200 m

You are standing 200 m away from the base of a building. You measure the top of the building to be at an angle of elevation (the angle between the ground and a line drawn to the top) of 60º. What is the height of the building?

Example

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Math Practice

This example (this slide and the next 2) addresses MP4 & MP5.

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SLIDE 25

cos(60º) sin(60º) 1 60º 60º height 200 m

Example

Make a quick sketch showing the original right triangle and

ne showing the appropriate trig

functions.

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cos(60 º) sin(60 º) 1 60 º height 200 m 60 º

Example

Then set up the ratios, substitute the values and solve.

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cos(60 º) sin(60 º) 1 60 º height 200 m 60 º

Example

Then set up the ratios, substitute the values and solve.

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Answer

height distance height = tanθ x distance h = tan(60º)(200m) h = (1.73)(200m) h = 346 m = tanθ

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SLIDE 26

9 You are standing 30 m away from the base of a

building. The top of the building lies at an angle of

elevation (the angle between the ground and the hypotenuse) of 50º. What is the height of the building? 50º height 30 m

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9 You are standing 30 m away from the base of a

building. The top of the building lies at an angle of

elevation (the angle between the ground and the hypotenuse) of 50º. What is the height of the building? 50º height 30 m

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Answer

36 m

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10 You are standing 50 m away from the base of a

building. The building creates an angle of elevation

with the ground measuring 80º. What is the height of the building?

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SLIDE 27

10 You are standing 50 m away from the base of a

building. The building creates an angle of elevation

with the ground measuring 80º. What is the height of the building?

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Answer

283 m

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11 Use the tanθ function of your calculator to determine the height of a flagpole if it is 30 m away and it's angle of elevation with the ground measures 70º.

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11 Use the tanθ function of your calculator to determine the height of a flagpole if it is 30 m away and it's angle of elevation with the ground measures 70º.

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Answer

82 m

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SLIDE 28

12 Use the tanθ function of your calculator to determine the height of a building if its base is 50 m away and it's angle of elevation with the ground measures 20º.

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12 Use the tanθ function of your calculator to determine the height of a building if its base is 50 m away and it's angle of elevation with the ground measures 20º.

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Answer

18 m

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13 You are on top of a building and look down to see someone who standing the ground. The angle of depression (the angle below the horizontal to an

bject) is 30º and they are 90 m from the base of

the building. How high is the building? (Neglect the heights of you and the other person.) Make sure to draw a sketch!

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SLIDE 29

13 You are on top of a building and look down to see someone who standing the ground. The angle of depression (the angle below the horizontal to an

bject) is 30º and they are 90 m from the base of

the building. How high is the building? (Neglect the heights of you and the other person.) Make sure to draw a sketch!

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Answer

52 m

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14 Determine the distance an object lies from the base

f a 45 m tall building if the angle of depression to

it is 40º.

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14 Determine the distance an object lies from the base

f a 45 m tall building if the angle of depression to

it is 40º.

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Answer

54 m

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SLIDE 30

When solving problems with trig, you find a right triangle which is similar to the one below. Then you find the solution by setting up the ratios of proportion. But, since the hypotenuse is 1, often it's forgotten that these are ratios. 1 cosθ sinθ θ

Trigonometric Ratios Slide 66 / 252

Trigonometric Ratios

Return to the Table of Contents

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Fill in the fundamental trig ratios below: Sine called "sin" for short Cosine called "cos" for short Tangent called "tan" for short

click click

Trigonometric Ratios

click

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SLIDE 31

Trigonometric Ratios

The name of the angle usually follows the trig function. If the angle is named θ (theta) the names become: · sinθ · cosθ · tanθ If the angle is named α (alpha) the functions become: · sinα · cosα · tanα

Slide 69 / 252 Trigonometric Ratios

If you have the sides, trig ratios let you find the angles. But if you have a side and an angle, trig ratios also let you find the other sides.

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These ratios depend on which angle you are calling θ; never the right angle. You know that the side

pposite the right angle is

called the hypotenuse. The leg opposite θ is called the opposite side. The leg that touches θ is called the adjacent side. hypotenuse adjacent side

pposite

side θ 1 cosθ sinθ θ

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SLIDE 32

Trigonometric Ratios

There are two possible angles that can be called #. Once you choose which angle is #, the names of the sides are defined. You can change later, but then the names of the sides also change. hypotenuse adjacent side θ

pposite side

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With this theta, these become the sides. 1 cosθ sinθ θ

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If you use the other angle, named α here, the names change accordingly. 1 cosα sinα α

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SLIDE 33

Trigonometric Ratios

hypotenuse adjacent side

pposite

side θ Let's say I'm solving a problem that involves this right triangle. To use trig, I'd find a right triangle with hypotenuse of 1 and legs of sinθ and cosθ which has the same angle θ so, it's similar.

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1 cosθ sinθ θ hypotenuse adjacent side

pposite

side θ Then set up the ratios. There are basic ratios relating the sides of these two triangles. Since they are similar triangles, the ratio of any two sides in one triangle is equal to that ratio of sides in the

ther.

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sinθ opposite side opp 1 hypotenuse hyp 1 cosθ sinθ θ hypotenuse adjacent side

pposite

side θ = =

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SLIDE 34

Trigonometric Ratios

cosθ adjacent side adj 1 hypotenuse hyp 1 cosθ sinθ θ hypotenuse adjacent side

pposite

side θ = =

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sinθ opposite side opp cosθ adjacent side adj 1 cosθ sinθ θ hypotenuse adjacent side

pposite

side θ = =

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sinθ opposite side opp 1 hypotenuse hyp cosθ adjacent side adj 1 hypotenuse hyp sinθ opposite side opp cosθ adjacent side adj 1 cosθ sinθ θ hypotenuse adjacent side

pposite

side θ = = = = = =

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SLIDE 35

Trigonometric Ratios

sinθ opposite side opp 1 hypotenuse hyp cosθ adjacent side adj 1 hypotenuse hyp sinθ opposite side opp cosθ adjacent side adj = = = = = = sinθ 1 = sinθ cosθ 1 = cosθ sinθ cosθ = tanθ But these can be simplified since:

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sinθ

pposite side

hypotenuse

pp

hyp adjacent side hypotenuse adj hyp cosθ tanθ

pposite side

adjacent side

pp

adj = = = = = = 1 cosθ sinθ θ hypotenuse adjacent side

pposite

side θ

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TOA SOH CAH

sinθ

pposite side

hypotenuse

pp

hyp adjacent side hypotenuse adj hyp cosθ tanθ

pposite side

adjacent side

pp

adj = = = = = = These trig ratios are used so often that they are memorized with the expression "SOH CAH TOA." If you get confused w/ the vowel sounds in SOH CAH TOA, you could also try the mnemonic sentence below. Some Old Horse Caught Another Horse Taking Oats Away.

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SLIDE 36

Trigonometric Ratios

TOA SOH CAH

sinθ

pposite side

hypotenuse

pp

hyp adjacent side hypotenuse adj hyp cosθ tanθ

pposite side

adjacent side

pp

adj = = = = = = These trig ratios are used so often that they are memorized with the expression "SOH CAH TOA." If you get confused w/ the vowel sounds in SOH CAH TOA, you could also try the mnemonic sentence below. Some Old Horse Caught Another Horse Taking Oats Away.

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Math Practice

MP6: Throughout this lesson, emphasize the correct set up of the trig ratios. Q's to help w/ this include: What side lengths am I given? (MP1) How are the sides related to the angle? (MP7) Which trig ratio am I going to use? (MP6) Construct the equation to solve this

problem. (MP2)

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15 Find the sinθ. Round your answer to the nearest hundredth. 3.0 8.5 θ 8.0

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15 Find the sinθ. Round your answer to the nearest hundredth. 3.0 8.5 θ 8.0

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Answer

0.94

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SLIDE 37

16 Find the cosθ. Round your answer to the nearest hundredth. 3.0 8.5 θ 8.0

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16 Find the cosθ. Round your answer to the nearest hundredth. 3.0 8.5 θ 8.0

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Answer

0.35

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17 Find the tanθ. Round your answer to the nearest hundredth. 3.0 8.5 θ 8.0

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SLIDE 38

17 Find the tanθ. Round your answer to the nearest hundredth. 3.0 8.5 θ 8.0

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Answer

2.67

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18 Find the tanθ. Round your answer to the nearest hundredth. 7 16 θ 14

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18 Find the tanθ. Round your answer to the nearest hundredth. 7 16 θ 14

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Answer

2

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SLIDE 39

19 Find the sinθ. Round your answer to the nearest hundredth. 7 16 θ 14

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19 Find the sinθ. Round your answer to the nearest hundredth. 7 16 θ 14

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Answer

0.88

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20 Find the cosθ. Round your answer to the nearest hundredth. 7 16 θ 14

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SLIDE 40

20 Find the cosθ. Round your answer to the nearest hundredth. 7 16 θ 14

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Answer

0.44

Slide 89 (Answer) / 252 Trigonometric Ratios

For instance, let's find the length of side x. The side we're looking for is

pposite the given angle;

and the given length is the hypotenuse; so we'll use the trig function that relates these three: 7.0 x 30º sinθ = =

pposite side

hypotenuse

pp

hyp

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7.0 x 30º sinθ = =

pposite side

hypotenuse

pp

hyp sinθ = opp hyp

pp = (hyp) (sinθ)

x = (7.0)(sin(30º)) x = (7.0)(0.50) x = 3.5

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SLIDE 41

Trigonometric Ratios

Now, let's find the length of side x in this case. The side we're looking for is adjacent the given angle; and the given length is the hypotenuse; so we'll use the trig function that relates these three: 9.0 x 25º adjacent side hypotenuse adj hyp cosθ = =

Slide 92 / 252 Trigonometric Ratios

9.0 x 25º adj = (hyp)(cosθ) x = (9.0)(cos(25º)) x = (9.0)(0.91) x = 8.2 adjacent side hypotenuse adj hyp cosθ = = adj hyp cosθ =

Slide 93 / 252 Trigonometric Ratios

Now, let's find the length of side x in this case. The side we're looking for is adjacent the given angle; and the given length is the

pposite the given angle;

so we'll use the trig function that relates these three: 9.0 x 50º tanθ =

pposite side

adjacent side = opp adj

Slide 94 / 252

SLIDE 42

Trigonometric Ratios

9.0 x 50º

pp = (adj)(tanθ)

x = (9.0)(tan(50º)) x = (9.0)(1.2) x = 10.8 tanθ = opposite side adjacent side= opp adj tanθ =

pp

adj

Slide 95 / 252

21 Find the value of x. Round your answer to the nearest tenth. 35 x 64º

Slide 96 / 252

21 Find the value of x. Round your answer to the nearest tenth. 35 x 64º

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Answer

17.1

Slide 96 (Answer) / 252

SLIDE 43

22 Find the value of x. Round your answer to the nearest tenth. 28 x 36º

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22 Find the value of x. Round your answer to the nearest tenth. 28 x 36º

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Answer

47.6

Slide 97 (Answer) / 252

23 Find the value of x. Round your answer to the nearest tenth. 28 x 44º

Slide 98 / 252

SLIDE 44

23 Find the value of x. Round your answer to the nearest tenth. 28 x 44º

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Answer

19.5

Slide 98 (Answer) / 252

24 Find the value of x. Round your answer to the nearest tenth. 7.4 x 37º

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24 Find the value of x. Round your answer to the nearest tenth. 7.4 x 37º

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Answer

5.9

Slide 99 (Answer) / 252

SLIDE 45

Applications of Trigonometric Ratios

Most of the time, trigonometric ratios are used to solve real- world problems, as you saw at the beginning of this unit. Now that you are familiar with the derivation of the three trigonometric ratios (sine, cosine, and tangent), you are ready to apply your knowledge and practice solving these problems. Before we begin, let's review some key vocabulary that you will see in these word problems.

Slide 100 / 252 Applications of Trigonometric Ratios

Most of the time, trigonometric ratios are used to solve real- world problems, as you saw at the beginning of this unit. Now that you are familiar with the derivation of the three trigonometric ratios (sine, cosine, and tangent), you are ready to apply your knowledge and practice solving these problems. Before we begin, let's review some key vocabulary that you will see in these word problems.

[This object is a pull tab]

Math Practice

These next 12 slides address MP1, MP2, MP4, MP5, MP6 & MP7. Use the questions provided in the "Math Practice" pull tab on slide #98 to provide additional assistance, when needed.

Slide 100 (Answer) / 252

angle of elevation sight line

bject

The angle of elevation is the angle above the horizontal to an object.

Applications of Trigonometric Ratios

The angle of depression is the angle below the horizontal to an object. angle of depression sight line

bserver
bject

Slide 101 / 252

SLIDE 46

20º 20º a n g l e

f

d e p r e s s i

n

angle of elevation 10,000ft The angle of elevation and the angle of depression are both measured relative to parallel horizontal lines, so they are equal in measure.

Applications of Trigonometric Ratios Slide 102 / 252 Applications of Trigonometric Ratios

Example Amy is flying a kite at an angle of 58º. The kite's string is 1 58 feet long and Amy's arm is 3 feet off the ground. How high is the kite off the ground? 58o 158 feet x 3 feet

Slide 103 / 252

sinθ = sin58 = .8480 = x = 134 x 158 x 158 x 158 Now, we must add in Amy's arm height. 134 + 3 = 137 The kite is about 137 feet off the ground. 158ft x 58º

Applications of Trigonometric Ratios Slide 104 / 252

SLIDE 47

30o 6 ft 5306 ft x Example You are standing on a mountain that is 5306 feet high. You look down at your campsite at angle of 30 º. If you are 6 feet tall, how far is the base of the mountain from the campsite?

Applications of Trigonometric Ratios Slide 105 / 252

30º x 5312 ft tan30 = .5774 = .5774x = 5312 x ≈ 9,200 ft 5312 x 5312 x The campsite is about 9,200 ft from the base of the mountain.

Applications of Trigonometric Ratios Slide 106 / 252

154 m Example: Vernon is on the top deck of a cruise ship and observes 2 dolphins following each other directly away from the ship in a straight line. Veron's position is 154 m above sea level, and the angles of depression to the 2 dolphins to the ship are 35 º and 36º, respectively. Find the distance between the 2 dolphins to the nearest hundredth of a meter.

Applications of Trigonometric Ratios Slide 107 / 252

SLIDE 48

154 m 35º x The first step is to divide the diagram into two separate ones. Then, find the horizontal distance in both. Let's call them x & y. 154 m 36º y Then, use your trigonometric ratios to find these values.

Applications of Trigonometric Ratios Slide 108 / 252

154 m 35º x tan 35 = 154 x 154 x 0.7002 = 0.7002x = 154 x = 219.94 m

Applications of Trigonometric Ratios Slide 109 / 252

154 m 36º y tan 36 = 154 y 154 y 0.7265 = 0.7265y = 154 y = 211.98 m

Applications of Trigonometric Ratios Slide 110 / 252

SLIDE 49

154 m 219.94 m 211.98 m Now, if we subtract these measurements, then we will find the distance between the 2 dolphins. 219.94 - 211.98 = 7.96 m

Applications of Trigonometric Ratios Slide 111 / 252

25 You are looking at the top of a tree. The angle of elevation is 55º. The distance from the top of the tree to your position (line of sight) is 84 feet. If you are 5.5 feet tall, how far are you from the base of the tree?

Slide 112 / 252

25 You are looking at the top of a tree. The angle of elevation is 55º. The distance from the top of the tree to your position (line of sight) is 84 feet. If you are 5.5 feet tall, how far are you from the base of the tree?

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Answer

55o 84 ft

x cos55 = x = 48.18 You are approximately 48 ft from the base of the tree. x 84

Slide 112 (Answer) / 252

SLIDE 50

26 A wheelchair ramp is 3 meters long and inclines at 6º. Find the height of the ramp to the nearest hundredth of a centimeter.

Slide 113 / 252

26 A wheelchair ramp is 3 meters long and inclines at 6º. Find the height of the ramp to the nearest hundredth of a centimeter.

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Answer

sin6 = y = 0.3136 m The height of the ramp is 31.36 cm y 3

6o

Slide 113 (Answer) / 252

27 John wants to find the height of a building which is casting a shadow of 175 ft at an angle of 73.75º. Find the height of the building to the nearest foot.

Slide 114 / 252

SLIDE 51

27 John wants to find the height of a building which is casting a shadow of 175 ft at an angle of 73.75º. Find the height of the building to the nearest foot.

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Answer

tan 73.75 = h = 600 The height of the building is 600 ft h 175 175 ft h 73.75o

Slide 114 (Answer) / 252

28 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. How deep is the submarine? 38° 800 m

Slide 115 / 252

28 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. How deep is the submarine? 38° 800 m

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Answer

492.53 meters

Slide 115 (Answer) / 252

SLIDE 52

29 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. If the submarine stays in the same position, then how far would the ship need to travel to be directly above the submarine? 38° 800 m

Slide 116 / 252

29 A sonar operator on a ship detects a submarine that is located 800 meters away from the ship at an angle of depression of 38º. If the submarine stays in the same position, then how far would the ship need to travel to be directly above the submarine? 38° 800 m

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Answer

630.41 meters

Slide 116 (Answer) / 252

30 The ship is traveling at a speed of 32 meters per second, in the direction towards the submarine. From its current position, how many minutes, to the nearest tenth of a minute, will it take the ship to be directly over the submarine. 38° 800 m

Slide 117 / 252

SLIDE 53

30 The ship is traveling at a speed of 32 meters per second, in the direction towards the submarine. From its current position, how many minutes, to the nearest tenth of a minute, will it take the ship to be directly over the submarine. 38° 800 m

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Answer

distance to get above the submarine is 630.41 meters (found in last question) 630.41/32 = 19.70 seconds 19.70/60 = 0.3 minutes

Slide 117 (Answer) / 252

Inverse Trigonometric Ratios

Return to the Table of Contents

Slide 118 / 252

So far, you have used the sine, cosine, and tangent ratios when given the measurement of the acute angle θ in a right triangle to find the measurements of the missing sides. What can you use when you need to find the measurements of the acute angles? We have what are called the inverse sine, inverse cosine and inverse tangent ratios that will help us answer the question above. If you know the measures of 2 sides of a triangle, then you can find the measurement of the angle with these ratios.

Inverse Trigonometric Ratios Slide 119 / 252

SLIDE 54

The Inverse Trigonometric Ratios are given below . If sinθ = , θ = sin-1 If cosθ = , θ = cos-1 If tanθ = , θ = tan-1

adj

pp

hyp A B C θ

Inverse Trigonometric Ratios

pp

hyp

( )

adj hyp

pp

adj

pp

hyp adj hyp

( )

pp

adj

( )

Remember:

S C T

h

a h

a

Slide 120 / 252

The Inverse Trigonometric Ratios are given below . If sinθ = , θ = sin-1 If cosθ = , θ = cos-1 If tanθ = , θ = tan-1

adj

pp

hyp A B C θ

Inverse Trigonometric Ratios

pp

hyp

( )

adj hyp

pp

adj

pp

hyp adj hyp

( )

pp

adj

( )

Remember:

S C T

h

a h

a

[This object is a pull tab]

Math Practice

MP6: Throughout this lesson, emphasize the correct set up of the trig ratios. Q's to help w/ this include: What side lengths am I given? (MP1) How are the sides related to the angle? (MP7) Which trig ratio am I going to use? (MP6) Construct the equation to solve this

problem. (MP2)

Slide 120 (Answer) / 252 Using Calculators with Inverse Trigonometry

The inverse trig functions are located just above the sine, cosine and tangent buttons. They are marked in the box on the calculator. On most calculators, they are noted by text which says SIN-1 COS-1 TAN-1 In most cases, they can be used by pressing the 2nd, or shift, button (arrow pointing to it) & the sine, cosine, or tangent button.

Slide 121 / 252

SLIDE 55

31 Find sin

1(0.8) Round the angle measure to the

nearest hundredth.

Slide 122 / 252

31 Find sin

1(0.8) Round the angle measure to the

nearest hundredth.

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Answer

θ = 53.13°

Slide 122 (Answer) / 252

32 Find tan

1(2.3). Round the angle measure to the

nearest hundredth.

Slide 123 / 252

SLIDE 56

32 Find tan

1(2.3). Round the angle measure to the

nearest hundredth.

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Answer

θ = 66.50°

Slide 123 (Answer) / 252

33 Find cos

1(0.45). Round the angle measurement to

the nearest hundredth.

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33 Find cos

1(0.45). Round the angle measurement to

the nearest hundredth.

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Answer

θ = 63.26°

Slide 124 (Answer) / 252

SLIDE 57

9 15 A B C θ To find an unknown angle measure in a right triangle, you need to identify the correct trig function that will find the missing value. Use "SOH CAH TOA" to help. ∠A is your angle of reference. Label the two given sides

f your triangle opp, adj, or hyp.

Identify the trig funtion that uses ∠A, and the two sides.

Inverse Trigonometric Ratios Slide 125 / 252

9 15 A B C θ hyp adj Using "SOH CAH TOA", I have "a" and "h", so the ratio is a/h which is cosine. now you can solve for m∠A, the missing angle using the inverse trig function.

Inverse Trigonometric Ratios

cos A = 9 15 m∠A = cos-1 9 15

( )

m∠A = 53.13º Once you find m∠A, you can easily find m∠C, using the Triangle Sum Theorem.

Slide 126 / 252 Inverse Trigonometric Ratios

Now, let's find the measurement of the angle θ in this case. The sides that we are given are the opposite side & the hypotenuse; so we'll use the trig function that relates these two sides with our angle: θ 12 13 sinθ = =

pposite side

hypotenuse

pp

hyp

Slide 127 / 252

SLIDE 58

Inverse Trigonometric Ratios

sin θ = 12 13 θ 12 13 θ = sin-1 12 13

( )

θ = 67.38º

Slide 128 / 252

34 Find the m∠D in the figure below. D E F 23 13

Slide 129 / 252

34 Find the m∠D in the figure below. D E F 23 13

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Answer

θ = 60.52°

Slide 129 (Answer) / 252

SLIDE 59

35 Find the m∠F in the figure below. D E F 27 35

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35 Find the m∠F in the figure below. D E F 27 35

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Answer

θ = 39.52°

Slide 130 (Answer) / 252

36 Find the m∠G in the figure below. G H J 17 18

Slide 131 / 252

SLIDE 60

36 Find the m∠G in the figure below. G H J 17 18

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Answer

θ = 70.81°

Slide 131 (Answer) / 252 Applications of Inverse Trigonometric Ratios

As we discussed earlier in this unit, trigonometric ratios and the inverse trigonometric ratios are used to solve real-world problems. Now that you are familiar with the three inverse trigonometric ratios (inverse sine, inverse cosine, and inverse tangent), you are ready to apply your knowledge and practice solving these problems.

Slide 132 / 252 Applications of Inverse Trigonometric Ratios

As we discussed earlier in this unit, trigonometric ratios and the inverse trigonometric ratios are used to solve real-world problems. Now that you are familiar with the three inverse trigonometric ratios (inverse sine, inverse cosine, and inverse tangent), you are ready to apply your knowledge and practice solving these problems.

[This object is a pull tab]

Math Practice

These next few examples (the next 4 slides) address MP1, MP2, MP4, MP5, MP6 & MP7. Use the questions provided in the "Math Practice" pull tab on slide #135 to provide additional assistance, when needed.

Slide 132 (Answer) / 252

SLIDE 61

Applications of Inverse Trigonometric Ratios

A hockey player is 24 feet from the goal line. He shoots the puck directly at the goal. The height of the goal is 4 feet. What is the maximum angle of elevation at which the player can shoot the puck and still score a goal? 4 ft 24 ft θ

Slide 133 / 252 Applications of Inverse Trigonometric Ratios

4 ft 24 ft θ tan θ = 4 24 θ = tan-1 4 24 θ = 9.46º The angle of elevation that the player can shoot the puck is a maximum of 9.46º.

( )

Slide 134 / 252 Applications of Inverse Trigonometric Ratios

You lean a 20 foot ladder up against a wall. The base of the ladder is 5 feet from the edge of the wall. What is the angle of elevation is created by the ladder & the ground. 20 ft 5 ft

Slide 135 / 252

SLIDE 62

Applications of Inverse Trigonometric Ratios

20 ft 5 ft cos θ = 5 20 θ = cos-1 5 20

( )

θ = 75.52º

Slide 136 / 252

37 Katherine looks down out of the crown of the statue

f liberty to an incoming ferry about 345 feet. The

distance from crown to the ground is about 250 feet. What is the angle of depression?

Slide 137 / 252

37 Katherine looks down out of the crown of the statue

f liberty to an incoming ferry about 345 feet. The

distance from crown to the ground is about 250 feet. What is the angle of depression?

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Answer

crown ferry 345 ft 250 ft The angle of depression is about 46 degrees.

Slide 137 (Answer) / 252

SLIDE 63

38 The Sear's Tower in Chicago, Illinois is 1451 feet tall. The sun is casting a 50 foot shadow on the ground. What is the angle of elevation created by the tip of the shadow and the ground? 1451 ft 50 ft

Slide 138 / 252

38 The Sear's Tower in Chicago, Illinois is 1451 feet tall. The sun is casting a 50 foot shadow on the ground. What is the angle of elevation created by the tip of the shadow and the ground? 1451 ft 50 ft

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Answer

θ = 88.03°

Slide 138 (Answer) / 252

39 You lean a 30 foot ladder up against the side of your home to get into a bedroom on the second floor. The height of the window is 25 feet. What angle of elevation must you set the ladder at in order to reach the window? 30 ft 25 ft

Slide 139 / 252

SLIDE 64

39 You lean a 30 foot ladder up against the side of your home to get into a bedroom on the second floor. The height of the window is 25 feet. What angle of elevation must you set the ladder at in order to reach the window? 30 ft 25 ft

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Answer

θ = 56.44°

Slide 139 (Answer) / 252

40 You are looking out your bedroom window towards the tip of the shadow made by your home. Your friend measures the length of the shadow to be 10 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of your home's shadow.

Slide 140 / 252

40 You are looking out your bedroom window towards the tip of the shadow made by your home. Your friend measures the length of the shadow to be 10 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of your home's shadow.

[This object is a pull tab]

Answer θ = 63.43° θ θ 20 ft 10 ft

Slide 140 (Answer) / 252

SLIDE 65

41 You return to view your home's shadow 3 hours later. Your friend measures the length of the shadow to be 25 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of your home's shadow.

Slide 141 / 252

41 You return to view your home's shadow 3 hours later. Your friend measures the length of the shadow to be 25 feet long. If you are 20 feet off the ground, what is the angle of depression needed to see the tip of your home's shadow.

[This object is a pull tab]

Answer θ = 51.34° θ θ 20 ft 25 ft

Slide 141 (Answer) / 252

Review of the Pythagorean Theorem

Return to the Table of Contents

Slide 142 / 252

SLIDE 66

Review of Pythagorean Theorem

c2 = a2 + b2 "c" is the hypotenuse "a" and "b" are the two legs; which leg is "a" and which is "b" doesn't matter.

Slide 143 / 252

42 The legs of a right triangle are 7.0m and 3.0m, what is the length of the hypotenuse?

Slide 144 / 252

42 The legs of a right triangle are 7.0m and 3.0m, what is the length of the hypotenuse?

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Answer

7.6

Slide 144 (Answer) / 252

SLIDE 67

43 The legs of a right triangle are 2.0m and 12m, what is the length of the hypotenuse?

Slide 145 / 252

43 The legs of a right triangle are 2.0m and 12m, what is the length of the hypotenuse?

[This object is a pull tab]

Answer

12.2

Slide 145 (Answer) / 252

44 The hypotenuse of a right triangle has a length of 4.0m and one of its legs has a length of 2.5m. What is the length of the other leg?

Slide 146 / 252

SLIDE 68

44 The hypotenuse of a right triangle has a length of 4.0m and one of its legs has a length of 2.5m. What is the length of the other leg?

[This object is a pull tab]

Answer

3.1

Slide 146 (Answer) / 252

45 The hypotenuse of a right triangle has a length of 9.0m and one of its legs has a length of 4.5m. What is the length of the other leg?

Slide 147 / 252

45 The hypotenuse of a right triangle has a length of 9.0m and one of its legs has a length of 4.5m. What is the length of the other leg?

[This object is a pull tab]

Answer

7.8

Slide 147 (Answer) / 252

SLIDE 69

46 What is the length of the third side? 4 7

Slide 148 / 252

46 What is the length of the third side? 4 7

[This object is a pull tab]

Answer

8.1

Slide 148 (Answer) / 252

47 What is the length of the third side? 20 15

Slide 149 / 252

SLIDE 70

47 What is the length of the third side? 20 15

[This object is a pull tab]

Answer

25

Slide 149 (Answer) / 252

48 What is the length of the third side? 4 7

Slide 150 / 252

48 What is the length of the third side? 4 7

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Answer

5.7

Slide 150 (Answer) / 252

SLIDE 71

49 What is the length of the third side? 9 15

Slide 151 / 252

49 What is the length of the third side? 9 15

[This object is a pull tab]

Answer

12

Slide 151 (Answer) / 252

50 What is the length of the third side? 3 4

Slide 152 / 252

SLIDE 72

50 What is the length of the third side? 3 4

[This object is a pull tab]

Answer

5

Slide 152 (Answer) / 252

3 4 5

Triples are integer solutions

f the Pythagorean

Theorem. 3-4-5 is the most famous of the triples: You don't need a calculator if you recognize the sides are in this ratio.

Pythagorean Triples Slide 153 / 252

51 What is the length of the third side? 6 8

Slide 154 / 252

SLIDE 73

51 What is the length of the third side? 6 8

[This object is a pull tab]

Answer

10 Note: 6-8-10 = 2*(3-4-5)

Slide 154 (Answer) / 252

52 What is the length of the third side? 12 20

Slide 155 / 252

52 What is the length of the third side? 12 20

[This object is a pull tab]

Answer

16 Note: 12-16-20 = 4*(3-4-5)

Slide 155 (Answer) / 252

SLIDE 74

53 (sinθ)2 + (cosθ)2 = ? 1 sinθ cosθ

Slide 156 / 252

53 (sinθ)2 + (cosθ)2 = ? 1 sinθ cosθ

[This object is a pull tab]

Answer

12 = 1

Slide 156 (Answer) / 252

54 Katherine looks down out of the crown of the statue

f liberty to an incoming ferry about 345 feet. The

distance from crown to the ground is about 250 feet. What is the distance from the ferry to the base of the statue?

Slide 157 / 252

SLIDE 75

54 Katherine looks down out of the crown of the statue

f liberty to an incoming ferry about 345 feet. The

distance from crown to the ground is about 250 feet. What is the distance from the ferry to the base of the statue?

[This object is a pull tab]

Answer

crown ferry 250 ft base

f the statue

345 ft The ferry is about 238 feet away from the statue.

x

Slide 157 (Answer) / 252

Converse of the Pythagorean Theorem

Return to the Table of Contents

Slide 158 / 252

If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. If c2 = a2 + b2, then ΔABC is a right triangle. a b c A B C

Converse of the Pythagorean Theorem Slide 159 / 252

SLIDE 76

Example

Tell whether the triangle is a right triangle . Explain your reasoning.

Remember c is the longest side D E F 7 24 25

Slide 160 / 252 Example

Tell whether the triangle is a right triangle . Explain your reasoning.

Remember c is the longest side D E F 7 24 25

[This object is a pull tab]

Answer

If 252 = 242 + 72, then ΔABC is a right Δ. If c2 = a2 + b2, then ΔABC is a right Δ. therefore ΔABC is a right Δ. 625 = 576 + 49 625 = 625 This example addresses MP2 & MP3 Additional Q's to address MP standards: What information do you have? (MP1) How do you determine if the triangle is a right triangle? (MP3) Construct an equation to solve the problem (MP2)

Slide 160 (Answer) / 252

If the square of the longest side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is obtuse. If c2 > a2 + b2, then ΔABC is obtuse. A B C a b c

Theorem Slide 161 / 252

SLIDE 77

If the square of the longest side of a triangle is less than the sum of the squares of the other two sides, then the triangle is acute. If c2 < a2 + b2, then ΔABC is acute. a b c A B C

Theorem Slide 162 / 252 Example

Classify the triangle as acute, right, or obtuse. Explain your reasoning.

17 15 13

Slide 163 / 252 Example

Classify the triangle as acute, right, or obtuse. Explain your reasoning.

17 15 13

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Answer

c = 17 Since c2 < a2 + b2 the triangle is acute. 172 ? 152 + 132 289 ? 225 + 169 289 < 394

Slide 163 (Answer) / 252

SLIDE 78

55 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 11 12 15

Slide 164 / 252

55 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 11 12 15

[This object is a pull tab]

Answer

A

Slide 164 (Answer) / 252

56 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 6 4 10

Slide 165 / 252

SLIDE 79

56 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 6 4 10

[This object is a pull tab]

Answer

D

Slide 165 (Answer) / 252

57 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 6 3 5

Slide 166 / 252

57 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 6 3 5

[This object is a pull tab]

Answer

C

Slide 166 (Answer) / 252

SLIDE 80

58 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 25 19 20

Slide 167 / 252

58 Classify the triangle as acute, right, obtuse, or not a triangle. A acute B right C

btuse

D not a triangle 25 19 20

[This object is a pull tab]

Answer

A

Slide 167 (Answer) / 252

59 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. A acute B right C

btuse

Slide 168 / 252

SLIDE 81

59 Tell whether the lengths 35, 65, and 56 represent the sides of an acute, right, or obtuse triangle. A acute B right C

btuse

[This object is a pull tab]

Answer

A

Slide 168 (Answer) / 252

60 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. A acute triangle B right triangle C

btuse triangle

Slide 169 / 252

60 Tell whether the lengths represent the sides of an acute, right, or obtuse triangle. A acute triangle B right triangle C

btuse triangle

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Answer

C

Slide 169 (Answer) / 252

SLIDE 82

Review

If c2 = a2 + b2, then triangle is right. If c2 < a2 + b2, then triangle is acute. If c2 > a2 + b2, then triangle is obtuse.

Slide 170 / 252

Special Right Triangles

Return to the Table of Contents

Slide 171 / 252

In this section you will learn about the properties of the two special right triangles. 45o 45o 90o 30o 60o 90o 45-45-90 30-60-90

Special Right Triangles Slide 172 / 252

SLIDE 83

Investigation: 45-45-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

1 1 C 45º 2 2 y 45º

Slide 173 / 252 Investigation: 45-45-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

1 1 C 45º 2 2 y 45º

2 2 y 45º

[This object is a pull tab]

Answer

c = √2 y = 2√2

Slide 173 (Answer) / 252 Investigation: 45-45-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

3 3 W 45º 4 4 C 45º

Slide 174 / 252

SLIDE 84

Investigation: 45-45-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

3 3 W 45º 4 4 C 45º

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Answer

w = 3√2 c = 4√2

Slide 174 (Answer) / 252 Investigation: 45-45-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

5 5 C 45º 6 6 x 45º

Slide 175 / 252 Investigation: 45-45-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

5 5 C 45º 6 6 x 45º

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Answer

c = 5√2 x = 6√2

Slide 175 (Answer) / 252

SLIDE 85

45-45-90 Triangle Theorem

Using the side lengths that you found in the Investigation, can you figure out the rule, or formula, for the 45-45-90 Triangle Theorem?

Slide 176 / 252 45-45-90 Triangle Theorem

Using the side lengths that you found in the Investigation, can you figure out the rule, or formula, for the 45-45-90 Triangle Theorem?

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Answer A 45-45-90 triangle is an isosceles right triangle, where the hypotenuse is √2 times the length of the leg.

45o 45o x√2

x x Math Practices: MP7 & MP8

Slide 176 (Answer) / 252

45º x√2 x x 45º

45-45-90 Triangle Theorem

This theorem can be proved algebraically using Pythagorean Theorem. a2 + b2 = c2 x2 + x2 = c2 2x2 = c2 x√2 = c

Slide 177 / 252

SLIDE 86

Find the length of the missing sides. Write the answer in simplest radical form.

45-45-90 Example

45º y x 45º 6 P R Q

Slide 178 / 252

Find the length of the missing sides. Write the answer in simplest radical form.

45-45-90 Example

45º y x 45º 6 P R Q

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Answer By the Corollary to the Base Angles Thm, PQ=QR. y=6 hypotenuse = √2(leg) x = √2(6) x = 6√2

Slide 178 (Answer) / 252

Find the length of the missing sides. Write the answer in simplest radical form.

45-45-90 Example

x 18 y S V T

Slide 179 / 252

SLIDE 87

Find the length of the missing sides. Write the answer in simplest radical form.

45-45-90 Example

x 18 y S V T

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Answer

Since, STU is an isosceles rt triangle

hypotenuse = leg

ST=TV x=y There are 2 ways to solve.

Slide 179 (Answer) / 252

Find the length of the missing sides. Write the answer in simplest radical form.

45-45-90 Example

x 8 y

Slide 180 / 252

Find the length of the missing sides. Write the answer in simplest radical form.

45-45-90 Example

x 8 y

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Answer

Slide 180 (Answer) / 252

SLIDE 88

61 Find the value of x. A 5 B 5√2 C 5√2 2 x 5 y

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61 Find the value of x. A 5 B 5√2 C 5√2 2 x 5 y

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Answer

C

Slide 181 (Answer) / 252

62 What is the length of the hypotenuse of an isosceles right triangle if the length of the legs is 8 √2 inches.

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SLIDE 89

62 What is the length of the hypotenuse of an isosceles right triangle if the length of the legs is 8 √2 inches.

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Answer hypotenuse = leg( )

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63 What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm.

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63 What is the length of each leg of an isosceles, if the length of the hypotenuse is 20 cm.

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Answer hypotenuse = leg( )

Slide 183 (Answer) / 252

SLIDE 90

Investigation: 30-60-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

z 2 1 30º 60º y 2 4 30º 60º

Slide 184 / 252 Investigation: 30-60-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

z 2 1 30º 60º y 2 4 30º 60º

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Answer

z = √3 y = 2√3

Slide 184 (Answer) / 252 Investigation: 30-60-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

w 3 6 30º 60º v 4 8 30º 60º

Slide 185 / 252

SLIDE 91

Investigation: 30-60-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

w 3 6 30º 60º v 4 8 30º 60º

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Answer

w = 3√3 c = 4√3

Slide 185 (Answer) / 252 Investigation: 30-60-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

u 5 10 30º 60º t 6 12 30º 60º

Slide 186 / 252 Investigation: 30-60-90 Triangle Theorem

Find the missing side lengths in the triangles. Leave answers in simplified radical/fractional form...NO DECIMALS!

u 5 10 30º 60º t 6 12 30º 60º

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Answer

u = 5√3 t = 6√3

Slide 186 (Answer) / 252

SLIDE 92

30-60-90 Triangle Theorem

Using the side lengths that you found in the Investigation, can you figure out the rule, or formula, for the 30-60-90 Triangle Theorem?

Slide 187 / 252 30-60-90 Triangle Theorem

Using the side lengths that you found in the Investigation, can you figure out the rule, or formula, for the 30-60-90 Triangle Theorem?

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Answer In a 30-60-90 right triangle, the hypotenuse is twice the length of the shorter leg and the longer leg is √3 times the length of the shorter leg.

x x# 3 2x 30o 60o

Math Practices: MP7 & MP8

Slide 187 (Answer) / 252

This theorem can be proved using an equilateral triangle and Pythagorean Theorem. x x√3 2x 30o 60o A C 60º 60º 30º 30º c=2x 2x a=x x D b B For right triangle ABD, BD is a perpendicular bisector. let a = x, c = 2x and b = BD a2 + b2 = c2 x2 + b2 = (2x)2 x2 + b2 = 4x2 b2 = 3x2 b = x√3

30-60-90 Triangle Theorem Slide 188 / 252

SLIDE 93

Example: Find the length of the missing sides

f the right triangle.

30-60-90 Example

y 5 x 30º 60º G H F

Slide 189 / 252

Recall triangle inequality, the shortest side is

pposite the smallest angle and the longest

side is opposite the largest angle. HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF

y 5 x 30º 60º G H F

30-60-90 Example Slide 190 / 252

Recall triangle inequality, the shortest side is

pposite the smallest angle and the longest

side is opposite the largest angle. HF is the shortest side GF is the longest side (hypotenuse) GH is the 2nd longest side HF < GH < GF

y 5 x 30º 60º G H F

30-60-90 Example

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Answer hypotenuse = 2(shorter leg) x = 2(5) x = 10 longer leg = √3(shorter leg) y = √3(5) y = 5√3

Slide 190 (Answer) / 252

SLIDE 94

Example: Find the length of the missing sides

f the right triangle.

y 9 x 30º 60º M A T

30-60-90 Example Slide 191 / 252

Example: Find the length of the missing sides

f the right triangle.

y 9 x 30º 60º M A T

30-60-90 Example

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Answer longer leg = √3(shorter leg) 9 = √3(x) 3√3 = x MA is the shorter leg and MT is the longer leg

Slide 191 (Answer) / 252

Example: Find the length of the missing sides of the right triangle.

30-60-90 Example

y 15 x 30º 60º

Slide 192 / 252

SLIDE 95

Example: Find the length of the missing sides of the right triangle.

30-60-90 Example

y 15 x 30º 60º

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Answer

Slide 192 (Answer) / 252

Example: Find the area of the triangle.

14 ft

30-60-90 Example Slide 193 / 252

Example: Find the area of the triangle.

14 ft

30-60-90 Example

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Math Practice

These next 2 example (slides #207 - 209) address MP1, MP2, MP4, MP5. Additional Q's to address MP's: What information do you have? (MP1) What side length(s) can you find? (MP1) What should we add to our diagram to solve this problem? (MP4 & MP5) Create an equation for this problem. (MP2)

Slide 193 (Answer) / 252

SLIDE 96

14 ft h

? ?

The altitude (or height) divides the triangle into two 30o-60o-90o triangles. The length of the shorter leg is 7 ft. The length of the longer leg is 7√3 ft. A = 1/2 b(h) = 1/2 14(7√3) A = 49√3 square ft ≈ 84.87 square ft

30-60-90 Example Slide 194 / 252

30o 9 ft

Example: Find the area of the triangle.

30-60-90 Example Slide 195 / 252

30o 9 ft

Example: Find the area of the triangle.

30-60-90 Example

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Answer

short leg = 4.5 ft long leg = 4.5 ft

Slide 195 (Answer) / 252

SLIDE 97

64 Find the value of x. A 7 B 7√3 C D 14 7 x 30º 60º 7√2 2

Slide 196 / 252

64 Find the value of x. A 7 B 7√3 C D 14 7 x 30º 60º 7√2 2

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Answer

B

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65 Find the value of x. A 7 B 7√3 C D 14 7√2 x 7√2 2

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SLIDE 98

65 Find the value of x. A 7 B 7√3 C D 14 7√2 x 7√2 2

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Answer

D

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66 Find the value of x. A 7 B 7√3 C D 14 30o 60o x 7√3 7√2 2

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66 Find the value of x. A 7 B 7√3 C D 14 30o 60o x 7√3 7√2 2

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Answer

A

Slide 198 (Answer) / 252

SLIDE 99

67 The hypotenuse of a 30º-60º-90º triangle is 13 cm. What is the length of the shorter leg?

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67 The hypotenuse of a 30º-60º-90º triangle is 13 cm. What is the length of the shorter leg?

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Answer shorter leg = 13/2 shorter leg = 6.5cm

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68 The length the longer leg of a 30º-60º-90º triangle is 7

cm. What is the length of the hypotenuse?

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SLIDE 100

68 The length the longer leg of a 30º-60º-90º triangle is 7

cm. What is the length of the hypotenuse?

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Answer shorter leg = hypotenuse = =

Slide 200 (Answer) / 252

The wheelchair ramp at your school has a height of 2.5 feet and rises at angle of 30º. What is the length of the ramp?

Real World Example Slide 201 / 252

The wheelchair ramp at your school has a height of 2.5 feet and rises at angle of 30º. What is the length of the ramp?

Real World Example

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Math Practice

This example (this and the next 2 slides) addresses MP1, MP2, MP4 & MP5. Additional Q's to address MP's: What information do you have? (MP1) What side length(s) can you find? (MP1) What should we add to our diagram to solve this problem? (MP4 & MP5) Create an equation for this problem. (MP2)

Slide 201 (Answer) / 252

SLIDE 101

30o 2.5 ? The triangle formed by the ramp is a 30º-60º-90º right

triangle. The length of the ramp is the hypotenuse.

hypotenuse = 2(shorter leg) hypotenuse = 2(2.5) hypotenuse = 5 The ramp is 5 feet long.

Real World Example Slide 202 / 252

69 A skateboarder constructs a ramp using plywood. The length of the plywood is 3 feet long and falls at an angle of 45º. What is the height of the ramp? Round to the nearest hundredth. 45o 3 feet ?

Slide 203 / 252

69 A skateboarder constructs a ramp using plywood. The length of the plywood is 3 feet long and falls at an angle of 45º. What is the height of the ramp? Round to the nearest hundredth. 45o 3 feet ?

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Answer

Slide 203 (Answer) / 252

SLIDE 102

70 What is the length of the base of the ramp? Round to the nearest hundredth. 45o 3 feet ?

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70 What is the length of the base of the ramp? Round to the nearest hundredth. 45o 3 feet ?

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Answer

Slide 204 (Answer) / 252

71 The yield sign is shaped like an equilateral triangle. Find the length of the altitude. 20 inches

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SLIDE 103

Slide 205 (Answer) / 252

72 The yield sign is shaped like an equilateral

triangle. Find the area of the sign.

20 inches

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72 The yield sign is shaped like an equilateral

triangle. Find the area of the sign.

20 inches

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Answer

Slide 206 (Answer) / 252

SLIDE 104

PARCC Sample Questions

The remaining slides in this presentation contain questions from the PARCC Sample Test. After finishing this unit, you should be able to answer these questions. Good Luck! Return to Table

f Contents

Slide 207 / 252

Question 10/25

An archaeological team is excavating artifacts from a sunken merchant vessel on the ocean floor. To help with teh exploration the team uses a robotic probe. The probe travels approximately 3,900 meters at an angle of depression of 67.4 degrees from the team's ship on the ocean surface down to the sunken vessel on the ocean

floor. The figure shows a representation of the team's sip and the

probe. Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 208 / 252

73 When the probe reaches the ocean floor, the probe will be approximately __________ meters below the ocean surface. A 1,247 B 1,500 C 1,623 D 3,377 E 3,600 Question 10/25

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 209 / 252

SLIDE 105

73 When the probe reaches the ocean floor, the probe will be approximately __________ meters below the ocean surface. A 1,247 B 1,500 C 1,623 D 3,377 E 3,600 Question 10/25

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

E

Slide 209 (Answer) / 252

74 When the probe reaches the ocean floor, the horizontal distance of the probe behind the team's ship on the

cean surface will be approximately ___________

meters. A B C D E F 1,247 G 1,500 H 1,623 I 3,377 J 3,600 Question 10/25

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 210 / 252

74 When the probe reaches the ocean floor, the horizontal distance of the probe behind the team's ship on the

cean surface will be approximately ___________

meters. A B C D E F 1,247 G 1,500 H 1,623 I 3,377 J 3,600 Question 10/25

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

G

Slide 210 (Answer) / 252

SLIDE 106

75 In right triangle ABC, m∠B ≠ m∠C. Let sin B = r and cos B = s. What is sin C - cos C? A r + s B r - s C s - r s D r B A C Question 3/25

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 211 / 252

75 In right triangle ABC, m∠B ≠ m∠C. Let sin B = r and cos B = s. What is sin C - cos C? A r + s B r - s C s - r s D r B A C Question 3/25

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

C

Slide 211 (Answer) / 252

An unmanned aerial vehicle (UAV) is equipped with cameras used to monitor forest fires. The figure represents a moment in time at which a UAV, at point B, flying at an altitude of 1,000 meters (m) is directly above point D on the forest floor. Point A represents the location

f a small fire on the forest floor.

Question 16/25

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 212 / 252

SLIDE 107

76 At the moment in time represented by the figure, the angle of depression from the UAV to the fire has a measure of 30º. At the moment in time represented by the figure, what is the distance from the UAV to the fire? Question 16/25 Part A

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 213 / 252

76 At the moment in time represented by the figure, the angle of depression from the UAV to the fire has a measure of 30º. At the moment in time represented by the figure, what is the distance from the UAV to the fire? Question 16/25 Part A

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

2000 meters

Slide 213 (Answer) / 252

77 What is the distance, to the nearest meter, from the fire to point D? Question 16/25 Part B

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 214 / 252

SLIDE 108

77 What is the distance, to the nearest meter, from the fire to point D? Question 16/25 Part B

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

1732 meters

Slide 214 (Answer) / 252

78 Points C and E represent the linear range of view of the camera when it is pointed directly down at point D. The field of view of the camera is 20º and is represented in the figure by ∠CBE. The camera takes a picture directly

ver point D, what is the approximate width of the forest

floor that will be captured in the picture? A 170 meters B 353 meters C 364 meters D 728 meters Question 16/25 Part C

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 215 / 252

78 Points C and E represent the linear range of view of the camera when it is pointed directly down at point D. The field of view of the camera is 20º and is represented in the figure by ∠CBE. The camera takes a picture directly

ver point D, what is the approximate width of the forest

floor that will be captured in the picture? A 170 meters B 353 meters C 364 meters D 728 meters Question 16/25 Part C

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

B

Slide 215 (Answer) / 252

SLIDE 109

79 The UAV is flying at a speed of 13 meters per second in the direction toward the fire. Suppose the altitude of the UAV is now 800 meters. The new position is represented at F in the figure. From its position at point F, how many minutes, to the nearest tenth of a minute, will it take the UAV to be directly over the fire? A 0.6 B 1.2 C 1.8 D 2.0 Question 16/25 Part D

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 216 / 252

79 The UAV is flying at a speed of 13 meters per second in the direction toward the fire. Suppose the altitude of the UAV is now 800 meters. The new position is represented at F in the figure. From its position at point F, how many minutes, to the nearest tenth of a minute, will it take the UAV to be directly over the fire? A 0.6 B 1.2 C 1.8 D 2.0 Question 16/25 Part D

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

C

Slide 216 (Answer) / 252

A spring is attached at one end to support B and at the other end to collar A, as represented in the figure. Collar A slides along the vertical bar between points C and D. In the figure, the angle θ is the angle created as the collar moves between points C and D. 80 When θ = 28°, what is the distance from point A to point B to the nearest tenth of a foot?

Question 20/25 Part A

Topic: Trigonometric Ratios PARCC Released Question (EOY)

Slide 217 / 252

SLIDE 110

A spring is attached at one end to support B and at the other end to collar A, as represented in the figure. Collar A slides along the vertical bar between points C and D. In the figure, the angle θ is the angle created as the collar moves between points C and D. 80 When θ = 28°, what is the distance from point A to point B to the nearest tenth of a foot?

Question 20/25 Part A

Topic: Trigonometric Ratios PARCC Released Question (EOY)

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Answer

3.4 feet

Slide 217 (Answer) / 252

81 When the spring is stretched and the distance from A to B is 5.2 feet, what is the value of θ to the nearest tenth of a degree? A 35.2° B 45.1° C 54.8° D 60.0° A spring is attached at one end to support B and at the other end to collar A, as represented in the figure. Collar A slides along the vertical bar between points C and D. In the figure, the angle θ is the angle created as the collar moves between points C and D.

Question 20/25 Part B

Topic: Inverse Trigonometric Ratios PARCC Released Question (EOY)

Slide 218 / 252

81 When the spring is stretched and the distance from A to B is 5.2 feet, what is the value of θ to the nearest tenth of a degree? A 35.2° B 45.1° C 54.8° D 60.0° A spring is attached at one end to support B and at the other end to collar A, as represented in the figure. Collar A slides along the vertical bar between points C and D. In the figure, the angle θ is the angle created as the collar moves between points C and D.

Question 20/25 Part B

Topic: Inverse Trigonometric Ratios PARCC Released Question (EOY)

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Answer

C

Slide 218 (Answer) / 252

SLIDE 111

82 Right triangle WXY is similar to triangle DEF. The following are measurements in right triangle DEF. A 90 B √113 C 7 D 8 m∠F = 90º DE = √113 DF = 7 EF = 8 Write an expression that represents cos W. Which number represents the numerator of the fraction? PARCC Released Question (PBA)

Question 4/7

Topic: Trigonometric Ratios

Slide 219 / 252

82 Right triangle WXY is similar to triangle DEF. The following are measurements in right triangle DEF. A 90 B √113 C 7 D 8 m∠F = 90º DE = √113 DF = 7 EF = 8 Write an expression that represents cos W. Which number represents the numerator of the fraction? PARCC Released Question (PBA)

Question 4/7

Topic: Trigonometric Ratios

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Answer

C

Slide 219 (Answer) / 252

83 Right triangle WXY is similar to triangle DEF. The following are measurements in right triangle DEF. A 90 B √113 C 7 D 8 m∠F = 90º DE = √113 DF = 7 EF = 8 Write an expression that represents cos W. Which number represents the denominator of the fraction? PARCC Released Question (PBA)

Question 4/7

Topic: Trigonometric Ratios

Slide 220 / 252

SLIDE 112

83 Right triangle WXY is similar to triangle DEF. The following are measurements in right triangle DEF. A 90 B √113 C 7 D 8 m∠F = 90º DE = √113 DF = 7 EF = 8 Write an expression that represents cos W. Which number represents the denominator of the fraction? PARCC Released Question (PBA)

Question 4/7

Topic: Trigonometric Ratios

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Answer

B

Slide 220 (Answer) / 252

84 The degree measure of an angle in a right triangle is x, and sin x = 1/3. Which of these expressions are also equal to 1/3? Select all that apply. A cos(x) B cos(x - 45°) C cos(45° - x) D cos(60° - x) E cos(90° - x) Question 6/7

Topic: Trigonometric Ratios PARCC Released Question (PBA)

Slide 221 / 252

84 The degree measure of an angle in a right triangle is x, and sin x = 1/3. Which of these expressions are also equal to 1/3? Select all that apply. A cos(x) B cos(x - 45°) C cos(45° - x) D cos(60° - x) E cos(90° - x) Question 6/7

Topic: Trigonometric Ratios PARCC Released Question (PBA)

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Answer

E

Slide 221 (Answer) / 252

SLIDE 113

85 In this figure, triangle GHJ is similar to triangle PQR. Based on this information, which ratio represents tan H? A B C D 8 15 8 17 15 8 17 8 Question 7/7

Topic: Trigonometric Ratios PARCC Released Question (PBA)

J G H R P Q 8 15 17

Slide 222 / 252

85 In this figure, triangle GHJ is similar to triangle PQR. Based on this information, which ratio represents tan H? A B C D 8 15 8 17 15 8 17 8 Question 7/7

Topic: Trigonometric Ratios PARCC Released Question (PBA)

J G H R P Q 8 15 17

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Answer

A

Slide 222 (Answer) / 252

86 Mariela is standing in a building and looking out of a window at a

tree. The tree is 20 feet away from Mariela. Mariela's line of sight

to the top of the tree creates a 42° angle of elevation, and her line

f sight to the base of the tree creates a 31° angle of depression.

What is the height, in feet, of the tree? Type in your answer.

Question 5/11

Topic: Trigonometric Ratios PARCC Released Question (PBA)

Slide 223 / 252

SLIDE 114

86 Mariela is standing in a building and looking out of a window at a

tree. The tree is 20 feet away from Mariela. Mariela's line of sight

to the top of the tree creates a 42° angle of elevation, and her line

f sight to the base of the tree creates a 31° angle of depression.

What is the height, in feet, of the tree? Type in your answer.

Question 5/11

Topic: Trigonometric Ratios PARCC Released Question (PBA)

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Answer

tan(42) = x 20 x = 20tan(42) x = 18 or 18.01 tan(31) = y 20 y = 20tan(31) y = 12 or 12.02 Height of tree: 12 + 18 = 30 ft

r

12.02 + 18.01 = 30.03 ft

x 20 42° y 20 42°

x y

Slide 223 (Answer) / 252 Released PARCC Exam Question

The following question from the released PARCC - PBA exam uses what we just learned and combines it with what we learned earlier to create a good question. Please try it on your own. Then we'll go through the processes that we can use to solve it. PARCC Released Question (PBA)

Slide 224 / 252

Question 1/11

Topic: Trigonometric Ratios The figure shows the design of a shed that will be built. Use the figure to answer all parts of the task. The base of the shed will be a square measuring 18 feet by 18

feet. The height of the rectangular sides will be 9 feet. The

measure of the ansgle made by the roof with the side of the shed can vary and is labeled as x°. Different roof angles create different surface areas of the roof. The surface area of the roof will determine the number of roofing shingles needed in constructing the shed. To meet drainage requirements, the roof angles must be at least 117°. 9 feet 18 feet 1 8 f e e t x°

Slide 225 / 252

SLIDE 115

Without changing the measurements of the base of the shed, the builder is also considering using a roof angle that will create a roof surface area that is 10% less than the area obtained in Part A. Less surface area will require less roofing shingles. Will such an angle meet the specified drainage requirements. Explain how you came to your conclusion. The builder of the shed is considering using an angle that measures 125°. Determine the surface area of the roof if 125° angle is used. Explain or show your process. Part A Part B The roofing shingles cost $27.75 for a bundle. Each bundle can cover approximately 35 square feet. Shingles must be purchased in full bundles. The builder has a budget of $325 for shingles. What is the greatest angle the builder can use and stay within budget? Explain or show your process. Part C

Slide 226 / 252

87 What concepts could we use to solve this problem? A Area of a rectangle B Right Triangle Trigonometry C Angle Addition Postulate D All of the above Question 1/11 Part A

Topic: Trigonometric Ratios The builder of the shed is considering using an angle that measures 125°. Determine the surface area of the roof if 125° angle is used. Explain or show your process.

Slide 227 / 252

87 What concepts could we use to solve this problem? A Area of a rectangle B Right Triangle Trigonometry C Angle Addition Postulate D All of the above Question 1/11 Part A

Topic: Trigonometric Ratios The builder of the shed is considering using an angle that measures 125°. Determine the surface area of the roof if 125° angle is used. Explain or show your process.

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Answer

D All of the above

Slide 227 (Answer) / 252

SLIDE 116

88 If the value of x is 125°, what would be the m∠1? A 90° B 25° C 35° D 160° Question 1/11 Part A

Topic: Trigonometric Ratios

18 feet 9 feet

x°

1 2 Front view of the shed 9 feet 18 feet 1 8 f e e t x°

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88 If the value of x is 125°, what would be the m∠1? A 90° B 25° C 35° D 160° Question 1/11 Part A

Topic: Trigonometric Ratios

18 feet 9 feet

x°

1 2 Front view of the shed 9 feet 18 feet 1 8 f e e t x°

[This object is a pull tab]

Answer

C

Slide 228 (Answer) / 252

89 What would be the value of y in the figure to the right? A 6 ft B 9 ft C 12 ft D 18 ft Question 1/11 Part A

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x° 18 feet 9 feet

x°

1 2 Front view of the shed

y

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SLIDE 117

89 What would be the value of y in the figure to the right? A 6 ft B 9 ft C 12 ft D 18 ft Question 1/11 Part A

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x° 18 feet 9 feet

x°

1 2 Front view of the shed

y

[This object is a pull tab]

Answer

B

Slide 229 (Answer) / 252

90 What ratio would we use to find the value of z in the figure below? A sin(35) = B tan(35) = C cos(35) = D tan(35) = Question 1/11 Part A

Topic: Trigonometric Ratios

z

18 feet 9 feet

x°

1 2 Front view of the shed

y

z 18 9 z 9 z z 9

9 feet 18 feet 1 8 f e e t x°

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90 What ratio would we use to find the value of z in the figure below? A sin(35) = B tan(35) = C cos(35) = D tan(35) = Question 1/11 Part A

Topic: Trigonometric Ratios

z

18 feet 9 feet

x°

1 2 Front view of the shed

y

z 18 9 z 9 z z 9

9 feet 18 feet 1 8 f e e t x°

[This object is a pull tab]

Answer

C

Slide 230 (Answer) / 252

SLIDE 118

91 What is the value of z in the figure below? A 7.37 feet B 10.32 feet C 10.99 feet D 12.85 feet Question 1/11 Part A

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

y

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91 What is the value of z in the figure below? A 7.37 feet B 10.32 feet C 10.99 feet D 12.85 feet Question 1/11 Part A

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

y

[This object is a pull tab]

Answer

C

Slide 231 (Answer) / 252

92 What is the area of the roof? A 98.91 ft2 B 197.82 ft2 C 296.73 ft2 D 395.64 ft2 Question 1/11 Part A

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

y

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SLIDE 119

92 What is the area of the roof? A 98.91 ft2 B 197.82 ft2 C 296.73 ft2 D 395.64 ft2 Question 1/11 Part A

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

y

[This object is a pull tab]

Answer

D If you got B as your answer, you forgot to multiply by 2 (2 rectangles that make up the roof)

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Question 1/11 Part B

Topic: Trigonometric Ratios Without changing the measurements of the base of the shed, the builder is also considering using a roof angle that will create a roof surface area that is 10% less than the area obtained in Part A. Less surface area will require less roofing shingles. Will such an angle meet the specified drainage requirements. Explain how you came to your conclusion.

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93 After finding the answer that the area of the roof was 395.64 ft2, what would be the area of a roof that has 10% less area? A 356.08 ft2 B 316.52 ft2 C 197.8 ft2 D 39.56 ft2 Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

Slide 234 / 252

SLIDE 120

93 After finding the answer that the area of the roof was 395.64 ft2, what would be the area of a roof that has 10% less area? A 356.08 ft2 B 316.52 ft2 C 197.8 ft2 D 39.56 ft2 Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

A If you got D as your answer, that represents the 10% of roofing removed, not the area of the new roof (need to subtract your answer from 395.64).

Slide 234 (Answer) / 252

94 Using the area that we found in the previous slide, what is the new value of z? A 10.99 ft B 17.58 ft C 19.78 ft D 9.89 ft Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

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94 Using the area that we found in the previous slide, what is the new value of z? A 10.99 ft B 17.58 ft C 19.78 ft D 9.89 ft Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

D 356.08 2(18) = 9.89 there are 2 rectangles that form the roof (2 in front of the 18 in equation above).

Slide 235 (Answer) / 252

SLIDE 121

95 Using the new value of z, what is the new m∠1? A 65.51° B 24.49° C 42.30° D 47.70° Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

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95 Using the new value of z, what is the new m∠1? A 65.51° B 24.49° C 42.30° D 47.70° Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

B 9 9.89 m∠1 = cos-1 ( )

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96 Does the measurement of our new angle x meet the building requirements? Yes No Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

Slide 237 / 252

SLIDE 122

96 Does the measurement of our new angle x meet the building requirements? Yes No Question 1/11 Part B

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

No, 114.49° is less than the 125° angle requirement.

Slide 237 (Answer) / 252

Question 1/11 Part C

Topic: Trigonometric Ratios The roofing shingles cost $27.75 for a bundle. Each bundle can cover approximately 35 square feet. Shingles must be purchased in full bundles. The builder has a budget of $325 for shingles. What is the greatest angle the builder can use and stay within budget? Explain or show your process.

Slide 238 / 252

97 If the roofing shingles cost $27.75 for a bundle and his budget is $325, how many bundles of shingles can he buy? A 10 B 11 C 11.71 D 12 Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

Slide 239 / 252

SLIDE 123

97 If the roofing shingles cost $27.75 for a bundle and his budget is $325, how many bundles of shingles can he buy? A 10 B 11 C 11.71 D 12 Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

B

Slide 239 (Answer) / 252

98 If each bundle of shingles covers an area of 35 square feet, then what is the area is covered by the the amount of bundles that the builder purchased? A 420 ft2 B 409.85 ft2 C 385 ft2 D 350 ft2 Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

Slide 240 / 252

98 If each bundle of shingles covers an area of 35 square feet, then what is the area is covered by the the amount of bundles that the builder purchased? A 420 ft2 B 409.85 ft2 C 385 ft2 D 350 ft2 Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

C

Slide 240 (Answer) / 252

SLIDE 124

99 Using the new area found in the last question, what is the value of z in the figures below? A 10.69 ft B 14.26 ft C 16.04 ft D 21.39 ft Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

Slide 241 / 252

99 Using the new area found in the last question, what is the value of z in the figures below? A 10.69 ft B 14.26 ft C 16.04 ft D 21.39 ft Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

A

Slide 241 (Answer) / 252

100 Using the new value of z found in the last question, what is the new value of x in the figures below? A 32.66° B 57.34° C 122.66° D 147.34° Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

Slide 242 / 252

SLIDE 125

100 Using the new value of z found in the last question, what is the new value of x in the figures below? A 32.66° B 57.34° C 122.66° D 147.34° Question 1/11 Part C

Topic: Trigonometric Ratios

9 feet 18 feet 1 8 f e e t x°

z z

18 feet 9 feet

x°

1 2 Front view of the shed

9 ft

[This object is a pull tab]

Answer

C If your answer was A, then you forgot to add 90° to your answer.

Slide 242 (Answer) / 252 Released PARCC Exam Question

The following question from the released PARCC - PBA exam uses what we just learned and combines it with what we learned earlier to create an interesting question. Please try it on your own. Then we'll go through the processes that we can use to solve it.

Slide 243 / 252

A billboard at ground level has a support length of 26 feet that extends from the top of the billboard to the ground. A post that is 5 feet tall is attached to the support and is 4 feet from where the base

f the support is attached to the ground. In the figure shown, the

distance, in feet, from the base of the billboard to the base of the support is labeled x. Create an equation that can be used to determine x. Discuss any assumptions that should be made concerning the equation. Use your equation to find the value of x. Show your work or explain your answer.

Question 3/11

Topic: Problem Solving w/ Similar Triangles

Slide 244 / 252

SLIDE 126

101 Is this problem solvable? Yes No Question 3/11

Topic: Problem Solving w/ Similar Triangles

Slide 245 / 252

101 Is this problem solvable? Yes No Question 3/11

Topic: Problem Solving w/ Similar Triangles

[This object is a pull tab]

Answer

5 4 x (b) 26 y Your answer to this question could go either way. Because they want you to "create an equation" and "discuss assumptions", the graders want you to say, "Yes" and explain the assumptions that you are

making. If you redraw the picture

and label it w/ the information that was provided, then you can see that this problem is not solvable, at first, because there are too many unknown values. If you assume that both the billboard and the post are perpendicular w/ the ground, then we have some right triangle tools that we could apply. 5 4 x (b) 26 y

5 b 4 x y 26 = =

Slide 245 (Answer) / 252

5 4 x (b) 26 y 102 If we assume that both the billboard & the post are perpendicular with the ground, what concepts could we use to solve this problem? A Pythagorean Theorem B Right Triangle Trigonometry C Similar Triangles D All of the above Question 3/11

Topic: Problem Solving w/ Similar Triangles

Slide 246 / 252

SLIDE 127

5 4 x (b) 26 y 102 If we assume that both the billboard & the post are perpendicular with the ground, what concepts could we use to solve this problem? A Pythagorean Theorem B Right Triangle Trigonometry C Similar Triangles D All of the above Question 3/11

Topic: Problem Solving w/ Similar Triangles

[This object is a pull tab]

Answer

D Note: choice C has to be used in combination w/ A and/or B. We'll show how the problem is solved using 2 methods.

Slide 246 (Answer) / 252

5 4 x (b) 26 y 103 What would be the value of y? A 3 B 9 C √41 D 41

Let's use first, the combination of A Pythagorean Theorem & C Similar Triangles.

Question 3/11

Topic: Problem Solving w/ Similar Triangles

Slide 247 / 252

5 4 x (b) 26 y 103 What would be the value of y? A 3 B 9 C √41 D 41

Let's use first, the combination of A Pythagorean Theorem & C Similar Triangles.

Question 3/11

Topic: Problem Solving w/ Similar Triangles

[This object is a pull tab]

Answer

C

Slide 247 (Answer) / 252

SLIDE 128

104 What proportion would we use to find the value of x? A B C D 5 4 x (b) 26 y 4 x √41 26 = 5 b √41 26 = 5 x √41 26 = 4 x 26 √41 = Question 3/11

Topic: Problem Solving w/ Similar Triangles

Slide 248 / 252

104 What proportion would we use to find the value of x? A B C D 5 4 x (b) 26 y 4 x √41 26 = 5 b √41 26 = 5 x √41 26 = 4 x 26 √41 = Question 3/11

Topic: Problem Solving w/ Similar Triangles

[This object is a pull tab]

Answer

B

Slide 248 (Answer) / 252

105 What is the value of x? 5 4 x (b) 26 y Question 3/11

Topic: Problem Solving w/ Similar Triangles

Slide 249 / 252

SLIDE 129

105 What is the value of x? 5 4 x (b) 26 y Question 3/11

Topic: Problem Solving w/ Similar Triangles

[This object is a pull tab]

Answer

104 = √41x x = 104/√41 x ≈ 16.24 feet 4 x √41 26 =

Slide 249 (Answer) / 252

5 4 x (b) 26 y G E F 106 What would be the ratio that we would use to find the measurement of Angle G? A B C D

Now, let's use the combination of B Right Triangle Trigonomety & C Similar Triangles.

Question 3/11

Topic: Problem Solving w/ Similar Triangles

5 26 sin G = 4 26 cos G = 5 4 tan G = 4 5 tan G =

Slide 250 / 252

5 4 x (b) 26 y G E F 106 What would be the ratio that we would use to find the measurement of Angle G? A B C D

Now, let's use the combination of B Right Triangle Trigonomety & C Similar Triangles.

Question 3/11

Topic: Problem Solving w/ Similar Triangles

5 26 sin G = 4 26 cos G = 5 4 tan G = 4 5 tan G =

[This object is a pull tab]

Answer

A

Slide 250 (Answer) / 252

SLIDE 130

5 4 x (b) 26 y G E F 107 What is the measurement of angle G? Question 3/11

Topic: Problem Solving w/ Similar Triangles

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5 4 x (b) 26 y G E F 107 What is the measurement of angle G? Question 3/11

Topic: Problem Solving w/ Similar Triangles

[This object is a pull tab]

Answer

m∠G = tan-1 m∠G = 51.34° 5 4

( )

5 4 tan G =

Slide 251 (Answer) / 252

5 4 x (b) 26 y G E F 108 Using the measurement of angle G, what is the value of x? Question 3/11

Topic: Problem Solving w/ Similar Triangles Since the two triangles are similar, the measurement of angle G is the same in both triangles.

Slide 252 / 252

SLIDE 131

5 4 x (b) 26 y G E F 108 Using the measurement of angle G, what is the value of x? Question 3/11

Topic: Problem Solving w/ Similar Triangles Since the two triangles are similar, the measurement of angle G is the same in both triangles.

[This object is a pull tab]

Answer

Slide 1 / 252

Geometry

Similar Triangles & Trigonometry

2015-10-22 www.njctl.org

Slide 2 / 252 Table of Contents

· Problem Solving with Similar Triangles · Trigonometric Ratios · Special Right Triangles · Review of the Pythagorean Theorem

· Similar Triangles and Trigonometry · PARCC Sample Questions · Inverse Trigonometric Ratios · Converse of the Pythagorean Theorem

Slide 3 / 252

Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

Slide 4 / 252

Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of

Slide 4 (Answer) / 252

Three basic approaches to real world problem solving include: · Similar Triangles · Trigonometry · Pythagorean Theorem

Problem Solving with Similar Triangles and Right Triangles Slide 5 / 252

Problem Solving with Similar Triangles

Return to the Table of Contents

Slide 6 / 252

Shadows and Similar Triangles Slide 7 / 252

Shadows and Similar Triangles

This example (this slides and the next 10) address MP1, MP2, MP3, MP4, MP5 & MP7

Slide 7 (Answer) / 252

When Thales visited the Great Pyramid of Giza 2600 years ago, it was already 2000 years old. He wanted to know its height.

Shadows and Similar Triangles Slide 8 / 252

Shadows and Similar Triangles Slide 9 / 252

What 2 facts can you recall from our study of similar triangles? Fill in the blanks below. Their angles are all conguent. Their corresponding sides are in proportion to one another.

Shadows and Similar Triangles

Slide 10 / 252

Shadows and Similar Right Triangles Slide 11 / 252 Shadows and Similar Right Triangles Slide 12 / 252 Shadows and Similar Right Triangles Slide 13 / 252

x

y x y Taking away the objects and just leaving the triangles created by the height of the object, the sunlight and the shadow on the ground, we can see these are similar triangles. All the angles are equal, so the sides must be in proportion.

Similar Right Triangles Slide 14 / 252

By putting one triangle atop the other it's easy to see that they are similar. Using the 2 ideas we came up with before, you know the angles are all the same, and the sides are in proportion.

Similar Right Triangles Slide 15 / 252

shadow

shadow of pyramid r a y s

Similar Right Triangles Slide 16 / 252

shadow

shadow of pyramid r a y s

Shadows and Similar Right Triangles Slide 17 / 252

2 m 1 m 120 m h height of pyramid height of rod pyramid's shadow rod's shadow =

Shadows and Similar Right Triangles

height of rod rod's shadow x pyramid's shadow height of pyramid = 1 m 2 m (120 m) h = h = 60 m

Slide 18 / 252

This approach can be used to measure the height of a lot of

And, a convenient measuring device is then your height, and the length of the shadow you cast. Try doing this on the next sunny day you can get outside. Measure the height of any object which is casting a shadow by comparing the length of its shadow to the length of your

Shadows and Similar Right Triangles

Lab - Indirect Measurement Reminder - Mirrors also create indirect measurement if you are doing this lab on a cloudy day.

Slide 19 / 252

This approach can be used to measure the height of a lot of

And, a convenient measuring device is then your height, and the length of the shadow you cast. Try doing this on the next sunny day you can get outside. Measure the height of any object which is casting a shadow by comparing the length of its shadow to the length of your

Shadows and Similar Right Triangles

Lab - Indirect Measurement Reminder - Mirrors also create indirect measurement if you are doing this lab on a cloudy day.

The lab addresses MP1, MP2, MP4, MP5 & MP7

Slide 19 (Answer) / 252

1 A lamppost casts a 9 ft shadow at the same time a person 6 ft tall casts a 4 ft shadow. Find the height of the lamppost. A 6 ft B 2.7 ft C 13.5 ft D 15 ft

Slide 20 / 252

1 A lamppost casts a 9 ft shadow at the same time a person 6 ft tall casts a 4 ft shadow. Find the height of the lamppost. A 6 ft B 2.7 ft C 13.5 ft D 15 ft

Answer

C Slide 20 (Answer) / 252

2 You're 6 feet tall and you notice that your shadow at one time is 3 feet long. The shadow

feet long How tall is the building?

Slide 21 / 252

2 You're 6 feet tall and you notice that your shadow at one time is 3 feet long. The shadow

feet long How tall is the building?

Answer

40 feet

Slide 21 (Answer) / 252

3 You're 1.5 m tall and you notice that your shadow at one time is 4.8 m long. The shadow

long How tall is the tree?

Slide 22 / 252

3 You're 1.5 m tall and you notice that your shadow at one time is 4.8 m long. The shadow

long How tall is the tree?

Answer

11 m

Slide 22 (Answer) / 252

4 Two buildings are side by side. The 35 m tall building casts a 21 m shadow. How long will the shadow of the 8 m tall building be at the same time?

Slide 23 / 252

4 Two buildings are side by side. The 35 m tall building casts a 21 m shadow. How long will the shadow of the 8 m tall building be at the same time?

Answer

4.8 m