Unit 6 Introduction to Trigonometry Degrees and Radians (Unit 6.2) - - PowerPoint PPT Presentation

Unit 6 – Introduction to Trigonometry Degrees and Radians (Unit 6.2)

William (Bill) Finch

Mathematics Department Denton High School

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds.

• W. Finch

DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds.

• W. Finch

DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds.

• W. Finch

DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds.

• W. Finch

DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds.

• W. Finch

DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Lesson Goals

When you have completed this lesson you will:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds.

• W. Finch

DHS Math Dept Radian/Degree 2 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Angles in Standard Position

An angle in standard position:

◮ starts on positive x-axis

(initial side)

◮ rotates counter-clockwise for

positive angles

◮ rotates clockwise for negative

angles

◮ often named with Greek letters

◮ theta . . . θ ◮ alpha . . . α ◮ beta . . . β

x y Initial Terminal Positive x y Initial Terminal Negative

• W. Finch

DHS Math Dept Radian/Degree 3 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Degree Measure

x (0◦) y 0◦ 30◦ 60◦ 90◦ 120◦ 150◦ 180◦ 210◦ 240◦ 270◦ 300◦ 330◦ 360◦ 45◦ 135◦ 225◦ 315◦

• W. Finch

DHS Math Dept Radian/Degree 4 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Degree-Minutes-Seconds (DMS)

A fraction of a degree can be expressed as a decimal fraction, but historically the degree was divided into minutes (′) and seconds (′′). 1◦ = 60′ and 1′ = 60′′ For example, 32.125◦ = 32◦ 7′ 30′′ Read “ 32 degrees, 7 minutes, and 30 seconds.”

• W. Finch

DHS Math Dept Radian/Degree 5 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Degree-Minutes-Seconds (DMS)

A fraction of a degree can be expressed as a decimal fraction, but historically the degree was divided into minutes (′) and seconds (′′). 1◦ = 60′ and 1′ = 60′′ For example, 32.125◦ = 32◦ 7′ 30′′ Read “ 32 degrees, 7 minutes, and 30 seconds.”

• W. Finch

DHS Math Dept Radian/Degree 5 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 1

Convert to decimal degrees. a) 25◦ 15′ b) 12◦ 10′ 33′′

• W. Finch

DHS Math Dept Radian/Degree 6 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Calculator Instructions – TI-84

• W. Finch

DHS Math Dept Radian/Degree 7 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 2

Convert to degree-minutes-seconds. a) 48.4◦ b) 21.456◦

• W. Finch

DHS Math Dept Radian/Degree 8 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Calculator Instructions – TI-84

• W. Finch

DHS Math Dept Radian/Degree 9 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r

• f the circle:

θ = s r where θ is measured in radians. x y r s r θ Note that in the diagram above the radius r of the circle is the same length as the arc s intercepted by the two radii, so θ = 1 rad when s = r.

• W. Finch

DHS Math Dept Radian/Degree 10 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

The circumference of a circle is

• ne revolution around the circle.

C = 2πr s = 2πr s r = 2π θ = 2π θ ≈ 6.28 x y θ A central angle θ that is one revolution is 2π radians.

• W. Finch

DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

The circumference of a circle is

• ne revolution around the circle.

C = 2πr s = 2πr s r = 2π θ = 2π θ ≈ 6.28 x y θ A central angle θ that is one revolution is 2π radians.

• W. Finch

DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

The circumference of a circle is

• ne revolution around the circle.

C = 2πr s = 2πr s r = 2π θ = 2π θ ≈ 6.28 x y θ A central angle θ that is one revolution is 2π radians.

• W. Finch

DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

The circumference of a circle is

• ne revolution around the circle.

C = 2πr s = 2πr s r = 2π θ = 2π θ ≈ 6.28 x y θ A central angle θ that is one revolution is 2π radians.

• W. Finch

DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

The circumference of a circle is

• ne revolution around the circle.

C = 2πr s = 2πr s r = 2π θ = 2π θ ≈ 6.28 x y θ A central angle θ that is one revolution is 2π radians.

• W. Finch

DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

The circumference of a circle is

• ne revolution around the circle.

C = 2πr s = 2πr s r = 2π θ = 2π θ ≈ 6.28 x y θ A central angle θ that is one revolution is 2π radians.

• W. Finch

DHS Math Dept Radian/Degree 11 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

One revolution around a circle is slightly more than 6 radians.

x y r 1 rad 2 rad 3 rad 4 rad 5 rad 6 rad s = r

• W. Finch

DHS Math Dept Radian/Degree 12 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

x y 0◦ 30◦ 60◦ 90◦ 120◦ 150◦ 180◦ 210◦ 240◦ 270◦ 300◦ 330◦ 360◦ 45◦ 135◦ 225◦ 315◦

π 6 π 4 π 3 π 2 2π 3 3π 4 5π 6

π

7π 6 5π 4 4π 3 3π 2 5π 3 7π 4 11π 6

2π

• W. Finch

DHS Math Dept Radian/Degree 13 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Special Angles – Learn Them!

x y π x y 2π x y π

2

x y

3π 2

• W. Finch

DHS Math Dept Radian/Degree 14 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Special Angles – Learn Them!

x y

π 4

x y

5π 4

x y

3π 4

x y

7π 4

• W. Finch

DHS Math Dept Radian/Degree 15 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Special Angles – Learn Them!

x y

π 3

x y

4π 3

x y

2π 3

x y

5π 3

• W. Finch

DHS Math Dept Radian/Degree 16 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Special Angles – Learn Them!

x y

π 6

x y

7π 6

x y

5π 6

x y

11π 6

• W. Finch

DHS Math Dept Radian/Degree 17 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian Measure

θ = π θ = 0 θ = 3π 2 θ = π 2 Quadrant I 0 < θ < π 2 (acute angles) Quadrant II π 2 < θ < π (obtuse angles) Quadrant III π < θ < 3π 2 Quadrant IV 3π 2 < θ < 2π

• W. Finch

DHS Math Dept Radian/Degree 18 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Radian-Degree Conversion

Set up and solve this proportion: radian degree = π rad 180◦ Hint – always set up the proportion with the unknown angle measure in the numerator.

• W. Finch

DHS Math Dept Radian/Degree 19 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 3

Convert to radian measure. a) 120◦ b) −30◦

• W. Finch

DHS Math Dept Radian/Degree 20 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 4

Convert to degree measure. a) −3π 4 b) 3π 2

• W. Finch

DHS Math Dept Radian/Degree 21 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Coterminal Angles

Coterminal angles have the same initial and terminal sides. x y α β x y α β To find a coterminal angle to some angle θ either add or subtract a multiple of 2π (or 360◦): θ ± n · 2π θ ± n · 360◦

• W. Finch

DHS Math Dept Radian/Degree 22 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Coterminal Angles

Coterminal angles have the same initial and terminal sides. x y α β x y α β To find a coterminal angle to some angle θ either add or subtract a multiple of 2π (or 360◦): θ ± n · 2π θ ± n · 360◦

• W. Finch

DHS Math Dept Radian/Degree 22 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 5

Sketch the angle given (in radians): θ = 2π 3 Then find two coterminal angles: one positive and one negative.

• W. Finch

DHS Math Dept Radian/Degree 23 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 6

Sketch the angle given (in radians): α = −π 4 Then find two coterminal angles: one positive and one negative.

• W. Finch

DHS Math Dept Radian/Degree 24 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 7

Sketch the angle given (in degrees): β = 25◦ Then find two coterminal angles: one positive and one negative.

• W. Finch

DHS Math Dept Radian/Degree 25 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 8

Sketch the angle given (in degrees): θ = −150◦ Then find two coterminal angles: one positive and one negative.

• W. Finch

DHS Math Dept Radian/Degree 26 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Arc Length

The relationship between a central angle and the length of the intercepted arc is s = rθ where θ is in radians. r s θ

• W. Finch

DHS Math Dept Radian/Degree 27 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 9

A circle has a radius of 5 inches. Find the length of the arc intercepted by a central angle of 120◦.

• W. Finch

DHS Math Dept Radian/Degree 28 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 10

Winnipeg, Manitoba (Canada) is approximately due north of

• Dallas. Winnipeg is at a latitude of 49◦ 53′ 0′′ N, and Dallas is

at a latitude of 32◦ 47′ 39′′ N. Use the given information to find the distance between Winnipeg and Dallas (assume the Earth is a perfect sphere with a radius of 4000 miles).

• W. Finch

DHS Math Dept Radian/Degree 29 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Area of a Sector

A sector of a circle is the region bounded by two radii and their intercepted arc. r θ The area of a sector is A = 1 2r 2θ (where θ is in radians).

• W. Finch

DHS Math Dept Radian/Degree 30 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 11

A sector has a radius of 12 inches and a central angle of 100◦. Find the area of the sector.

• W. Finch

DHS Math Dept Radian/Degree 31 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 12

Find the approximate area swept by the wiper blade shown, if the total length of the windshield wiper mechanism is 26 inches.

• W. Finch

DHS Math Dept Radian/Degree 32 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Linear and Angular Speed

An object moving along an arc has a linear speed given by ν = arc length time = s t An object moving along an arc has an angular speed given by ω = central angle time = θ t θ r s

• W. Finch

DHS Math Dept Radian/Degree 33 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Linear and Angular Speed

An object moving along an arc has a linear speed given by ν = arc length time = s t An object moving along an arc has an angular speed given by ω = central angle time = θ t θ r s

• W. Finch

DHS Math Dept Radian/Degree 33 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

Example 13

A bicycle wheel has a radius of 35 cm. A chalk mark is made on the tire and then the tire is spun completing one full revolution in 0.8 seconds. a) Determine the linear speed of the chalk mark. b) Determine the angular speed.

• W. Finch

DHS Math Dept Radian/Degree 34 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

What You Learned

You can now:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. ◮ Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,

39, 41, 43, 45, 51, 55, 57, 59

• W. Finch

DHS Math Dept Radian/Degree 35 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

What You Learned

You can now:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. ◮ Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,

39, 41, 43, 45, 51, 55, 57, 59

• W. Finch

DHS Math Dept Radian/Degree 35 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

What You Learned

You can now:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. ◮ Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,

39, 41, 43, 45, 51, 55, 57, 59

• W. Finch

DHS Math Dept Radian/Degree 35 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

What You Learned

You can now:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. ◮ Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,

39, 41, 43, 45, 51, 55, 57, 59

• W. Finch

DHS Math Dept Radian/Degree 35 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

What You Learned

You can now:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. ◮ Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,

39, 41, 43, 45, 51, 55, 57, 59

• W. Finch

DHS Math Dept Radian/Degree 35 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

What You Learned

You can now:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. ◮ Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,

39, 41, 43, 45, 51, 55, 57, 59

• W. Finch

DHS Math Dept Radian/Degree 35 / 35

Introduction Angles – Degrees Angles – Radians Coterminal Applications Summary

What You Learned

You can now:

◮ Understand an angle as a measure of rotation. ◮ Understand radian and degree measures. ◮ Be able to convert between radian and degree measure. ◮ Be able to calculate arc length and sector area. ◮ Be able to find angular and linear speeds. ◮ Do problems Chap 4.2 #1, 5, 11-25 odd, 29, 31, 33, 35,

39, 41, 43, 45, 51, 55, 57, 59

• W. Finch

DHS Math Dept Radian/Degree 35 / 35