chapter 13 sec 2 angles and degree measure. 2 of 21 algebra 2 chapter 13 sections 2 & 3 an angle...

Chapter 13 Sec 2

Angles and Degree Measure

2 of 21

Algebra 2 Chapter 13 Sections 2 & 3

• An angle in standard position has its vertex at the origin and initial side on the positive x–axis.

initial side

terminal side

Standard Position

3 of 21

Algebra 2 Chapter 13 Sections 2 & 3

• Angles that have a counterclockwise rotation have a positive measure.

130

0º

90º

180º

270º

Positively Counterclockwise

4 of 21

Algebra 2 Chapter 13 Sections 2 & 3

• Angles that have a clockwise rotation have a negative measure.

– 130

0º

– 90º

– 180º

– 270º

Clockwise means negative

5 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Now let’s look at angle measures 30, 150, 210, and 330.

180°(1, 0)(–1, 0)

30º30º

30º30º

30°150°

210°330°

They all form a 30° angle with the x-axis, so they should all have the same sine, cosine, and tangent values…only the signs will change!

The angle to the nearest x-axis is called the reference angle.

All angles with the same reference angle will have the same trig values except for sign changes.

Unit Circle

6 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Unit Circle• A unit circle is a circle with radius 1. • If we have an angle between 0o and 90o

in standard position. Let P(x, y) be the point of intersection. If a perpendicular segment is drawn we create a right triangle, where y is opposite θ and x is adjacent to θ.

• Right triangles can be formed for angles greater than 90o, simply use the reference angle.

yy

hyp

opp

1sin x

x

hyp

adj

1cos

7 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Radian…still

• A point P(x, y) is on the unit circle if and only if its distance from the origin is 1.

• The radian measure of an angle is the length of the corresponding arc on the unit circle.

• Since

P(x, y)s

α

radians 2360 thus,1 and 2 rrC

radians. 2

90 and radians 081 o... S

8 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Degree/Radian Conversion

degree180

Radians radians180

Degree

9 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Example 1a. Change 115o to radian measure in terms of π..

b. Change radian to degree measure.

180115 115 oo

36

23

8

7

180

8

7

8

7

5.157

10 of 21

Algebra 2 Chapter 13 Sections 2 & 3

30° and 45° Radians

• You will need to know these conversions.

11 of 21

Algebra 2 Chapter 13 Sections 2 & 3

• Coterminal angles are angles that have the same initial and terminal side, but differ by the number of rotations.

• Since one rotation equals 360, the measures of coterminal angles differ by multiples of 360.

300 – 360 =

300 60

60 + 360 = – 60420

Coterminal Angles

12 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Example 2Find one positive and one negative coterminal angle.

a. 45o

45o + 360o = 405o and 45o – 360 o = –315o

b. 225o

225o + 360o = 585o and 225o – 360 o = –135o

Chapter 13 Sec 3

Trigonometric Functions

14 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Radius other than 1.• Suppose we have a hypotenuse with a length

other than 1. For our example we’ll use r as the length.

• In standard position r extends from the Origin to point P(x, y).

15 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Quadrantal Angle• If a terminal side of an angle coincides with one

of the axes, the angle is called a quadrantal angle. See below for examples:

• A full rotation around the circle is 360o. Measures more than 360o represent multiple rotations.

16 of 21

Algebra 2 Chapter 13 Sections 2 & 3

To find the values of trig functions of angles greater than 90, you will need to know how to find the measures of the reference angle.

If θ in nonquadrantal, its reference angle is formed by the terminal side of the given angle and the x-axis.

Reference Angles

17 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Example 1Find the reference angle for each angle.

a. 312o Since 312o is between 270o and 360o the terminal

side is in fourth quad. Therefore, 360o – 312o = 48o.

b. –195o

the coterminal angle is 360o – 195o = 165o this put us in the second quadrant so… 180o – 165o = 15o

18 of 21

Algebra 2 Chapter 13 Sections 2 & 3

0°/360°

(1, 0)

90° (0, 1)

270° (0, –1)

180°

(–1, 0)

Students

Sine values are positive

(csc, too)

All

All values are positive

Take

Tangent values are positive

(cot, too)

Calculus

Cosine values are positive

(sec, too)

Determining Sign

19 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Example 2Find the values of the six trigonometric functions for angle θ in standard position if a point with coordinates (–15, 20) lies on the terminal side.

256252015 22 r

5

4

25

20sin

5

3

25

15cos

3

4

15

20tan

4

5

20

25csc

3

5

15

25sec

4

3

20

15cot

20 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Example 3Find the values of the six trigonometric functions

Suppose θ is an angle in standard position whose

terminal side lies in the Quadrant III. If

find the remaining five trigonometric functions of θ.

7

means III Quad

7

7

342

222

222

y

y

y

y

yxr

4

7sin

3

4sec

4

3cos

3

7

3

7tan

7

73

7

3cot

7

74

7

4csc

21 of 21

Algebra 2 Chapter 13 Sections 2 & 3

Daily Assignment• Chapter 13 Sections 2 & 3• Study Guide

• Pg 177• #4 – 7

• Pg 178 – 180 Odd