Chapter 13 Sec 2

Angles and Degree Measure

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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

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Algebra 2 Chapter 13 Sections 2 & 3

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

130

0º

90º

180º

270º

Positively Counterclockwise

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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

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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

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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

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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

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Algebra 2 Chapter 13 Sections 2 & 3

Degree/Radian Conversion

degree180

Radians radians180

Degree

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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

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Algebra 2 Chapter 13 Sections 2 & 3

30° and 45° Radians

• You will need to know these conversions.

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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

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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

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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).

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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.

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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

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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

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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

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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

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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

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Algebra 2 Chapter 13 Sections 2 & 3

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

• Pg 177• #4 – 7

• Pg 178 – 180 Odd