2. Chapter 8: The Unit Circle and the Functions of Trigonometry
8.1 Angles, Arcs, and Their Measures
8.2 The Unit Circle and Its Functions
8.3 Graphs of the Sine and Cosine Functions
8.4 Graphs of the Other Circular Functions
8.5 Functions of Angles and Fundamental Identities
8.6 Evaluating Trigonometric Functions
8.7 Applications of Right Triangles
8.8 Harmonic Motion

3. 8.6 Evaluating Trigonometric Functions
Acute angle A is drawn in
standard position as shown.
Right-Triangle-Based Definitions of Trigonometric Functions
For any acute angle A in standard position,

4. 8. 6. Finding Trigonometric Function Values
8.6 Finding Trigonometric Function Values of an Acute Angle in a Right Triangle
Example Find the values of sin A, cos A, and tan A
in the right triangle.
Solution
length of side opposite angle A is 7
length of side adjacent angle A is 24
length of hypotenuse is 25

5. 8.6 Trigonometric Function Values of Special Angles
Angles that deserve special study are 30º, 45º, and 60º.
Using the figures above, we have the exact values of the special angles summarized in the table on the right.

6. 8.6 Cofunction Identities
In a right triangle ABC, with right angle C, the acute angles A and B are complementary.
Since angles A and B are complementary, and
sin A = cos B, the functions sine and cosine are called cofunctions. Similarly for secant and cosecant, and tangent and cotangent.

7. 8.6 Cofunction Identities
If A is an acute angle measured in degrees, then
If A is an acute angle measured in radians, then
Note These identities actually apply to all angles (not just
acute angles).

8. 8.6 Writing Functions in Terms of Cofunctions
Example Find the reference angle for each angle.
cos 52º 16’ (b)
Solution
cos 52º16’ (b)
= sin(90º – 52º16’)
= sin 37º 44’ (b)

9. 8.6 Reference Angles
A reference angle for an angle , written  , is the positive acute angle made by the terminal side of angle  and the x-axis.
Example Find the reference angle for each angle.
218º (b)
Solution
(a)   = 218º – 180º = 38º (b)

10. 8.6 Special Angles as Reference Angles
Example Find the values of the trigonometric
functions for 210º.
Solution The reference angle for 210º is
210º – 180º = 30º.
Choose point P on the terminal side so that the distance from the origin to P is 2. A 30º - 60º right triangle is formed.

11. 8.6 Special Angles as Reference Angles
Finding Trigonometric Function Values for Any Nonquadrantal Angle
1. If then find a coterminal angle by adding or subtracting 360º as many times as needed to obtain an angle greater than 0º but less than 360º .
2. Find the reference angle
3. Find the trigonometric function values for reference angle
4. Determine the correct signs for the values found in Step 3. This give the values of the trigonometric functions for angle

12. 8.6 Finding Trigonometric Function Values Using Reference Angles
Example Find the exact value of each expression.
cos(–240º) (b) tan 675º
Solution
–240º is coterminal with 120º.
The reference angle is
180º – 120º = 60º. Since –240º
lies in quadrant II, the
cos(–240º) is negative.
Similarly,
tan 675º = tan 315º
= –tan 45º = –1.

13. 8.6 Finding Trigonometric Function Values with a Calculator
Example Approximate the value of each expression.
cos 49º 12 (b) csc º
Solution Set the calculator in degree mode.

14. 8.6 Finding Angle Measure
Example Using Inverse Trigonometric Functions to Find
Angles
Use a calculator to find an angle  in degrees that satisfies sin  
Use a calculator to find an angle  in radians that satisfies tan   0.25.
Solution
With the calculator in degree mode,
we find that an angle having a sine
value of is 75.4º. Write
this as sin  75.4º.
With the calculator in radian mode,
we find tan 

15. 8.6 Finding Angle Measure
Example Find all values of , if  is in the interval
[0º, 360º) and
Solution Since cosine is negative,  must lie in
either quadrant II or III. Since
So the reference angle   = 45º.
The quadrant II angle  = 180º – 45º = 135º, and the
quadrant III angle  = 180º + 45º = 225º.