Radians & Degrees Radians & Degrees & Co-terminal - - PDF document

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Algebra II

Trigonometric Functions

www.njctl.org 2015-12-17

Slide 3 / 162 Trig Functions

· Graphing · Radians & Degrees & Co-terminal angles · Unit Circle

click on the topic to go to that section

· Trigonometric Identities · Arc Length & Area of a Sector

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Radians & Degrees and Co-Terminal Angles

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A few definitions: A central angle of a circle is an angle whose vertex is the center of the circle. An intercepted arc is the part of the circle that includes the points of intersection with the central angle and all the points in the interior of the angle.

central angle intercepted arc

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Radians and Degrees One radian is the measure

• f a central angle that

intercepts an arc whose length is equal to the radius of the circle.

There are , or a little more than 6, radians in a circle.

Click on the circle for an animated view of radians.

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Converting from Degrees to Radians 360∘ = 2 radians 1∘ = = 2 360 180

radians

Example: Convert 50∘ and 90∘ to radians. 50∘ ⋅ 180 = 18 radians 90∘ ⋅ 180 = 2 radians 5 There are 360 in a circle. Therefore Use this conversion factor to covert degrees to radians. 5

radians

2 = 180 90∘

radians

18 = 180 50∘ Example: Convert 50∘ and 90∘ to radians.

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Converting from Radians to Degrees 2 radians = 360∘ 1 radian = = 2 180 Example: Convert and to radians 360 degrees 4 4 180

⋅

= 45∘

⋅

180 = 180∘ Use this conversion factor to covert radians to degrees.

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Converting between Radians and Degrees Convert degrees to radians

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Converting between Radians and Degrees Convert radians to degrees

radians radians radians Slide 11 / 162 Slide 12 / 162

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4 Convert radians to degrees:

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Angles

Initial side Terminal side Initial side Terminal side

Angle Angle in standard position

An angle is formed by rotating a ray about its endpoint. The starting position is the initial side and the ending position is the terminal side. When, on the coordinate plane, the vertex of the angle is the

• rigin and the initial side is the positive x-axis, the angle is in

standard position.

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Positive Angle - terminal side rotates in a counter- clockwise direction Negative Angle - terminal side rotates in a clockwise direction

α = - 37∘

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Drawing angles in standard position 310∘

40∘

Each quadrant is 90∘, and 310∘ is 40∘ more than 270∘, so the terminal side is 40∘ past the negative y-axis. 500∘ 500∘ is 140∘ more than 360∘, so the angle makes a complete revolution counterclockwise and then another 140∘.

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Coterminal Angles Angles that have the same terminating side are

• coterminal. To find coterminal angles add or subtract

multiples of 360∘ for degrees and 2 for radians. Example: Find one positive and one negative angle that are terminal with 75∘.

75∘

• 285∘

435∘

75 + 360 = 435 75 - 360 = -285

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5 Which angles are coterminal with 40∘? (Select all that are correct.)

A 320 B -320 C 400 D -400

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6 Which graph represents 425∘ ?

A B C D

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7 Which graph represents ?

A B C D

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8 Which angle is NOT coterminal with -55∘?

A 305∘ B 665∘ C -415∘ D -305∘

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9 Which angle is coterminal with ?

A B C D

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Arc Length & Area of a Sector

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Arc length: s = r Area of sector: A = Arc length and the area of a sector

r arc length s sector

How do these formulas relate to the area and the circumference of a circle?

(Measured in radians)

Area of sector: A = Arc length: s = r

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40∘ 45∘

Who is getting more pie? Who is getting more of the crust at the outer edge? Emily's slice is cut from a 9 inch pie. Chester's slice is cut from an 8 inch pie. (Assume both pies are the same height.)

(Try to work this out in your groups. The solution is on the next slide)

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40∘ 45∘

The top of Emily's piece has an area of The top of Chester's piece has an area of Emily's crust has a length of Chester's crust has a length of

click click

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10 What is the top surface area of this slice of pizza from an 18-inch pie?

45∘ Slide 29 / 162

11 What is the arc length of the outer edge of this slice of pizza from an 18-inch pie?

45∘ Slide 30 / 162

12 If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth?

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13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region?

8 inches

4 i n

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Unit Circle

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The circle x2 + y2 = 1, with center (0,0) and radius 1, is called the unit circle.

1

(1,0) (0,-1) (0,1) (-1,0)

Quadrant I: x and y are both positive Quadrant IV: x is positive and y is negative Quadrant III: x and y are both negative Quadrant II: x is negative and y is positive

The Unit Circle

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In this triangle, sin#= = b

θ a b (a,b) (1,0) (0,-1) (-1,0) (0,1)

b 1

cos# = = a

a 1

so the coordinates

• f (a,b) are also

(cos#, sin#)

1

The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures. For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point.

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In this example, the terminal point is in Quadrant IV.

For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ).

0.57 0.82

• 55

1

If we look at the triangle, we can see that sin(-55 ) = 0.82 cos(-55 ) = 0.57 EXCEPT that we have to take the direction into account, and so sin(-55 ) is negative because the y value is below the x-axis.

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Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems:

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1

In this example, we know the angle. Using a calculator, we find that cos 44∘ ≈ .72 and sin 44∘ ≈ .69, so the coordinates of C are approximately (0.72, 0.69). Note that 0.722 + 0.692 ≈ 1! What are the coordinates of point C?

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The Tangent Function Recall SOH-CAH-TOA

hypotenuse adjacent side

• pposite side

#

sin = #

• pp

hyp

cos = #

adj hyp

tan = #

• pp

adj

It is also true that tan = . # sin

cos

# #

Why?

• pp

hyp adj hyp

= ⋅ = = tan

• pp

hyp adj hyp

• pp

adj

#

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Angles in the Unit Circle Example: Given a terminal point on the unit circle (- ). Find the value of cos, sin and tan of the angle. Solution: Let the angle be . x = cos , so cos = . y = sin , so sin = . tan = = = =

⋅

(Shortcut: Just cross out the 41's in the complex fraction.)

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Example: Given a terminal point , find #, tan# and csc#.

To find #, use sin-1 or cos -1 : sin -1 ( ) = #

# # 28.1∘ tan# = sin#/ cos# tan # = csc# = 1/ sin# csc # =

Note the "hidden" Pythagorean Triple, 8, 15, 17).

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(__, - )

5 13

A

θ

22.3

• 22.3

Example: Find the x-value of point A, θ and the tan θ.

For every point on the circle, Since x is in quadrant III, x = -12

13

sin-1(- ) ≈ -22.3 , BUT θ is in quadrant III, so θ = 180 + 22.3 = 202.3 (notice how 202.3 and -22.3 have the same sine)

5 13

tan θ = = =

sin θ cos θ

5 12

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Example: Given the terminal point of (-5/13,-12/13). F ind sin x, cos x, and tan x.

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14 What is tan θ?

A B C D

(- ,__)

3 5 θ

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15 What is sin θ?

A B C D

(- ,__)

3 5 θ

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16 What is θ (give your answer to the nearest degree)?

(- ,__)

3 5 θ

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17 Given the terminal point , find tan x.

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Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45 . A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values.

Isosceles Right Triangle Equilateral Triangle (the altitude divides the triangle into two 30-60-90 triangles)

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Special Right Triangles

(see Triangle Trig Review unit for more detail on this topic)

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1 ( , )

45∘

Special Triangles and the Unit Circle

(- , )

• 1

45∘

Multiples of 45 angles have sin and cos of ± , depending on the quadrant.

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30o 45o 60o 30o 45o 60o 30o 45o 60o 30o 45o 60o

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Drag the degree and radian angle measures to the angles of the circle:

#

4 3# 4 5# 4 3# 2

#

2

#

7# 4 2#

0∘ 45∘ 90∘ 135∘ 180∘ 225∘ 270∘ 315∘ 360∘

Slide 53 / 162 #

4 3# 4 5# 4 3# 2

#

2

#

7# 4 2#

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

1

• 1

Fill in the coordinates of x and y for each point

• n the unit circle:

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30∘

( , )

60∘

( , )

1 1

Special Triangles and the Unit Circle Angles that are multiples of 30 have sin and cos

• f ± and ± .

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Drag the degree and radian angle measures to the angles of the circle:

#

6 2# 3 5# 3 3# 2

#

2 # 7# 6 2# 11# 6 4# 3 # 3 5# 6

0∘ 30∘ 60∘ 90∘ 120∘ 150∘ 180∘ 210∘ 240∘ 270∘ 300∘ 330∘ 360∘

Slide 56 / 162 #

6 2# 3 5# 3 3# 2

#

2

#

7# 6 2# 11# 6 4# 3 # 3 5# 6

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

Drag in the coordinates of x and y for each point

• n the unit circle:

( , ) ( , )

1

• 1

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Special Angles in Degrees

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Radian Values of Special Angles

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Exact Values of Special Angles

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Put it all together...

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Degrees 0∘ 30∘ 45∘ 60∘ 90∘ Radians sin θ cos θ tan θ Complete the table below: Exact values of special angles

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If we know one trig function value and the quadrant in which the angle lies, we can find the angle and the other trig values.

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• pp

adj hyp

• 3

Example: If tan = , and sin < 0, find sin , cos and the value of . Solution: Since tan is positive and sin is negative, the terminal side of must be in Quadrant III.

· Draw a right triangle in Quadrant III. · Use the Pythagorean Theorem to find the length of the hypotenuse:

(Continued on next slide)

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• pp

adj hyp

• 3

Once we know the lengths for each side, we can calculate the sin, cos and the angle. Used the signed numbers to get the correct values. sin = = cos = = Use any inverse trig function to find the

• angle. tan-1( ) ≈ 36.7 . Because the angle

is in QIII, we need to add 180 + 36.7 = 216.7, so ≈ 217 .

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22 Which functions are positive in the second quadrant? Choose all that apply.

A

cos x

B

sin x

C

tan x

D

sec x

E

csc x

F

cot x

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23 Which functions are positive in the fourth quadrant? Choose all that apply.

A

cos x

B

sin x

C

tan x

D

sec x

E

csc x

F

cot x

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24 Which functions are positive in the third quadrant? Choose all that apply.

A

cos x

B

sin x

C

tan x

D

sec x

E

csc x

F

cot x

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Graphing Trig Functions

Return to Table of Contents

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If you have Geogebra available on your computer, click the star below to download a geogebratube animated graph of the trig functions:

(Once the webpage opens, click on Download)

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Graphing the Sine Function, y = sin x Graph by creating a table of values of key points. One

• ption is to use the set of values for x that are multiples
• f , and the corresponding values of

y or sin x. (Remember, is just a bit more than 3.) Since the values are based on a circle, values will repeat.

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Notice that the graph of y = sin x increases from 0 to 1, then decreases back to 0 and then to -1 and then goes back up to 0, as x increases from 0 to 2 .

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Graphing the Cosine Curve

Since the values are based on a circle, values will repeat.

Make a table of values just as we did for sin. We could use any interval, but are choosing from 0 to 2 .

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Notice that the graph of y = cos x starts at 1, decreases to -1 and then goes back up to 1 as x increases from 0 to 2 .

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Compare the graphs: How are they similar and how are they different?

y = sin x

y = cos x

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Characteristics of y = sin x and y = cos x

Domain: set of real numbers (x can be anything) Range: -1 ≤ y ≤ 1 Amplitude: one-half the distance from the minimum of the graph to the maximum or 1. The functions are periodic - the pattern repeats every 2 units.

amplitude = 1 period = 2 range:

• 1 ≤ y ≤ 1

Slide 81 / 162 Predict, Explore, Confirm

• 1. Using your prior knowledge of transforming functions, predict what

happens to the following functions:

• 2. Using your graphing calculator, insert the parent function into and

the transformed function into . Compare the graphs.

• 3. Do your conclusions match your predictions?

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y = a sin x or y = a cos x Amplitude is a positive number that represents

• ne-half the difference between the minimum

and the maximum values, or the distance from the midline to the maximum.

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Consider the graphs of y = sin x y = 2sin x y = sin x

y = sin x y = 2sin x y = sin x

What do you notice about these graphs? What does the value of "a" do to the graph? Name the amplitude of each graph.

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As shown in the graph below, the graph of y = -3cos x is a reflection over the x-axis of the graph of y = 3cos x . What is the amplitude of each function?

y = 3cos x y = -3cos x

The domain of each function is the set of real numbers and the range is {x|-3 ≤ x ≤ 3}.

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Sketch each graph on the interval from 0 to 2 : y = 4cos x y = -.25 sin x

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25 What is the amplitude of y = 3cos x ?

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26 What is the amplitude of y = 0.25cos x ?

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27 What is the amplitude of y = -sin x ?

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28 What is the range of the function y = 2sin x ?

A All real numbers B -2 < x < 2 C 0 ≤ x ≤ 2 D -2 ≤ x ≤ 2

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29 What is the domain of y = -3cos x ?

A All real numbers B -3 < x < 3 C 0 ≤ x ≤ 3 D -3 ≤ x ≤ 3

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30 Which graph represents the function y = -2sin x ?

A B C D

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31 What is the amplitude of the graph below?

Slide 93 / 162 Predict, Explore, Confirm

• 1. Using your prior knowledge of transforming functions, predict what

happens to the following functions:

• 2. Using your graphing calculator, insert the parent function into and

the transformed function into . Compare the graphs.

• 3. Do your conclusions match your predictions?

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A periodic function is one that repeats its values at regular intervals. One complete repetition of the pattern is called a cycle. The period is the length of one complete cycle. The trig functions are periodic functions. The basic sine and cosine curves have a period of 2 , meaning that the graph completes one complete cycle in 2 units.

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y = sin bx or y = cos bx Consider the graphs of y = cos x and y = cos 2x .

y = cos x y = cos 2x

• ne cycle

Notice that the graph of y = cos 2x completes one cycle twice as fast, or in units.

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y = cos x completes 1 cycle in 2 #. So the period is 2π. y = cos 2x completes 2 cycles in 2 # or 1 cycle in #. The period is #. y = cos 0.5x completes a cycle in 4#. The period is 4#.

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The period for y = cos bx or y = sin bx is

2 b

y = cos x b = 1 P = = 2 y = cos 2x b = 2 P = = y = cos 0.5x b = 0.5 P = = 4

P =

2 1 2 2 2 0.5 2

b 2 Slide 98 / 162

32 What is the period of

A B C D

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33 What is the period of

A B C D

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34 What is the period of

A B C D

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Sketch the graph of each function from x = 0 to x = 2 .

y = 2cos 3x y = -2cos 2x y = sin 2x y = cos x

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35 What is the period of the graph below?

A B 2

C 3

D

2 2

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36 What is the period of the graph shown?

A B C D

2 3 2 3

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37 What is the equation of this function?

A B C D

y = sin 3x y = cos 3x y = 3cos x y = 3sin x

Slide 105 / 162 Predict, Explore, Confirm

• 1. Using your prior knowledge of transforming functions, predict what

happens to the following functions:

• 2. Using your graphing calculator, insert the parent function into and

the transformed function into . Compare the graphs.

• 3. Do your conclusions match your predictions?

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Translating Sine and Cosine Functions

Trig functions can be translated in the same way as any other function. The horizontal shift is called a phase shift. What are your conclusions from the graphing calculator activity?

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k

Horizontal or phase shift Vertical shift Drag each equation to the matching graph

y = cos x y = cos (x + )

2

y = sin x + 2 y = sin x

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Consider the graphs of In order to determine the phase shift, the coefficient of x must be factored out. In , the 2 is factored out. The phase shift is . In , when the 2 is factored out, we get . The phase shift is .

and

(which is which?)

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Another way to determine the phase shift is to set the quantity inside the parenthesis equal to 0 and solve for the variable.

Example: Set Solve for x: So, the phase shift is 2 .

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y= sin (x) + k or y= cos (x) + k

Vertical Shift The k moves the graph up or down.

The graph below is of the equation y = 2 sin (3x). The midline of this graph is the horizontal line y = 0. Sketch the graph of y = 2 sin (3x) + 1.

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42 What is the vertical shift in

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43 What is the vertical shift in

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44 What is the vertical shift in

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Graphing a Sine or Cosine Function:

Step 1: Identify the amplitude, period, phase shift

and vertical shift.

Step 2: Draw the midline (y = k) Step 3: Find 5 key points - maximums, minimums

and points on the midline

Step 4: Draw the graph through the 5 points.

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Example:

Step 1: Amplitude: |-1| = 1 Period: Phase Shift: Vertical Shift: 2 (up 2)

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Step 2: Draw the midline y = 2

Step 3: Find the 5 key points Note: for x, adding comes from dividing the cycle, 3 by 4. For y, adding and subtracting 1 comes from the amplitude.

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Step 4: Graph

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You try:

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49 What is the amplitude of this cosine graph?

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50 What is the period of this cosine graph? (use 3.14 for pi)

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51 What is the vertical shift of this cosine graph?

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52 Which of the following of the following is an equation for the graph?

A B C D

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The equation y = 4.2cos (π/6(x - 1)) + 13.7 can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, 1 - 12.

Sketch the graph of this equation. What is the average temperature in June?

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Graphing the Tangent Function

Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of , and the corresponding values of y or tan x.

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Notice that the tangent function is not defined for values of x where cos x = 0, or starting at and every units in either direction. This is shown on the graph by the vertical lines, or asymptotes at these x values. The period of the function is units, because there is one complete cycle from to . As x approaches

• r any odd multiple of from the left, y increases

and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity. This is shown on the graph by the vertical lines, or asymptotes at these x values. The period of the function is units, because there is one complete cycle from to . As x approaches

• r any odd multiple of from the left, y increases

and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity.

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Example: Sketch the graph of y = tan (x + ) + 2 Asymptotes will be at 0, , 2 , etc. The midline will be at y = 2.

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53 Which graph represents y = -tan x?

A B C D

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Trigonometric Identities

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Key Ideas An identity is a mathematical equation that is true for all defined values of the variable. A trigonometric identity is an identity that contains

• ne or more trig ratios.

By contrast, a conditional equation is one that is only true for a limited set of values. By learning trig identities, we will be able to replace complicated expressions with simpler ones to solve and verify more difficult equations and identities.

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Drag each equation into the correct box:

Identities Conditional Equations 3x + 4 = 9

3x + 4 = 3x + 4 sin # + cos # = 1

2x 5 =x3

tan θ cot θ =1

2(x-1) = 2x - 2

(x + 3)2 = x2 + 9 sin 4x = 4sin x

5x - 7y = -(7y - 5x)

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Basic Trig Identities

csc # = 1 sin # sec # = 1 cos # cot # = 1 tan # sin # = 1 csc # 1 1 cos # = 1 sec # tan # = 1 cot #

Reciprocal Identities Tangent Identity Cotangent Identity

tan # = sin # cos # cot # = cos # sin #

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By using the basic identities we can change an expression into an equivalent expression. Also, all of the rules of addition, subtraction, multiplication and division that we learned to solve equations and manipulate expressions can be applied to trig expressions and equations.

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Algebraic example Trig example

(x - y)(x + y) = x 2 - y

2

(1 - cos #)(1 + cos #) = 1 - cos

2#

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Recall the unit circle, x2 + y2 = 1.

1

(1,0) (0,-1) (0,1) (-1,0)

(cos #,sin #)

Pythagorean Identities

For any point (x, y) on the circle, its coordinates are (cos #, sin #). Therefore, (cos #)2 + (sin #)2 = 12, which is usually written as cos2 θ + sin2 θ = 1

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Pythagorean Identities

How do we transform the first identity, which is derived from the unit circle, to the other two?

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Alternative Forms of Identities Since we know that 3 + 5 = 8, we also know that 8 - 5 = 3 and 8 - 3 = 5. In elementary school we call these equivalent equations "fact families". Similarly, if cos2 θ + sin2 θ = 1, it follows that 1 - cos2 θ = sin2 θ and 1 - sin2 θ = cos2 θ.

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More Alternative Forms Another fact family tells that since = 4, it follows that 4 ⋅ 5 = 20. Since sec θ = , then sec θ cos θ = 1 (multiply both sides of the first equation by cos #).

20 5 1

cos θ

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Simplifying Trig Expressions Example 1: Simplify csc θ cos θ tan θ. Rewrite each trig ratio in terms of cos and sin: 1 sin #

sin θ ⋅ cos θ ⋅ = 1

cos #

(When multiplying fractions, it is

• ften easier to reduce or cancel

before you multiply.) Example 2: Simplify csc2 θ(1 - cos2 θ). 1

sin2 θ⋅ (sin2 θ) = 1

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Verifying an Identity Transform one side of the identity to be the same as the other side

sin # cot # = cos # sin # ⋅ = cos #

cos # sin #

Example 1: Verify

Example 2: Verify cos θ csc θ tan θ = 1 cos # ⋅ ⋅ = 1

1 sin # sin # cos #

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Simplify:

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Simplify:

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Simplify:

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Simplify:

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Verify:

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Verify:

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54 Which equation is NOT an identity?

A sin2 x= 1 - cos2 x B 2 cot x = C tan2 x = sec2 x - 1 D sin2 x = cos2 x - 1 2cos x sin x

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55 The following expression can be simplified to which choice?

A B C D

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56 The following expression can be simplified to which choice?

A B C D

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57 The following expression can be simplified to which choice?

A B C D