4.2 & 4.4: Trig Functions and The Unit Circle

Objectives:•Identify a unit circle and describe its relationship to real #’s•Evaluate trig functions using the unit circle•Use reference angles to evaluate trig functions for non-acute angles•Use domain and period to evaluate sine/cosine functions•Use a calculator to evaluate trig functions

THE UNIT CIRCLE

• Circle with a radius of 1: x2 + y 2 = 1• Used to evaluate trig functions

Each point on the unit circle (x,y) can also be used to find the 6 trig functions! This is huge!!

a. Draw a 60° angle in standard position.

b. Create a right triangle with the terminal side and the x-axis.

c. Find the other side lengths of the right triangle.

d. What is the sin (60°), cos (60°), tan (60°)?

e. What is the x coordinate on the unit circle? The y?

f. Notice anything???

This also works for angles that are greater than 90⁰. To do this we use reference angles

Let Ө be an angle in standard position. Its reference angle is the acute angle, Ө’, formed by the

terminal side of Ө and the horizontal axis The trig function’s value for Ө is the same as the associated

reference angle, Ө’

TO FIND REFERENCE ANGLES:Quadrant 2: Quadrant 3: Quadrant 4:

180 180

360

2

THE UNIT CIRCLE!!

Things to take notice of:x-coordinate is cos Ѳ, y-coordinate is sin ѲAn (x,y) ordered pair on the unit circle gives

you the sin and cos values, which will allow you to find other trig function values…AMAZING!!

Activity

In small groups, find the sin, cos, and tan of the following angles (WITHOUT YOUR BOOKS!):

Draw central angle in standard position, radius = 1Create a special right triangle with the terminal side and the x-axis Calculate the sin Ѳ, cos Ѳ, and tan Ѳ.

6

11,

3

5,

3

4,

6

7,

6

5,

3

2,6

Repeat with following angles:

4

7,

4

5,

4

3,4

On your unit circle, label and their (x,y) coordinates. Which of the trig functions are undefined at these angles?

2,

2

3,,

2,0

It’s Triggy

Getting Triggy With It!!

UNIT CIRCLE

Fill in the Unit Circle

Knowing the unit circle will help you tremendously. But you can always use special right triangles if you forget!

cos (-120°)

Find the sin, cos, and tan for each real number, t.

1.

2.

3.

4.

5.

6.

2

t

3

t

4

5t

4

7t

180t

4

9t

Definition of Trig Functions on The Unit Circle

t is a real number, (x,y) is the point on the unit circle corresponding to t:sin t = y csc t = 1/y , y ≠ 0cos t = x sec t = 1/x. x ≠ 0tan t = y/x, x≠0 cot t = x/y, y ≠ 0

Determine the exact values of the 6 trig functions.

(-8/17, 15/17)

DOMAIN and RANGE

Domain for sin and cos: All real numbers

Range:sin t = y cos t = x-1 < y < 1 -1 < x < 1

The sin and cos function values repeat after . They are called periodic functions.

Definition of Periodic Functions:A function, f, is periodic if there exists a positive real number c such that

f(t + c) = f(t) (the value of the functions are the same)

for all t in the domain of f. The least number c for which f is periodic is called the period of f.

(Think about it…. and have the same sin and cos values)

2

6

2

6

Examples: Evaluate

1. 2. 4

9sin

2

5cos

Sweet Website!!!

EVEN and ODD Trig Functions

Remember, even functions: f(-x) = f(x) odd functions: f(-x) = - f(x)

cos and sec are evencos (-t) = cos t sec(-t) = sec t

sin, csc, tan, cot are oddsin(-t) = -sin t, csc(-t) = -csc t, tan (-t) = -tan t, cot(-t) = -cot (t)

Examples

1. 2. Find sin (-t)= cos (-t)=

csc (-t)= sec (-t)=

3csc;3

1sin tt

5

1cos t

Using what you know about the unit circle, why does it make sense that sin2θ + cos2 θ =1?