1. Finding Exact Values For Trigonometry Functions (Then Using those Values to Evaluate Trigonometry functions and Solve Trigonometry Equations)

2. Review: Special Right Triangles
Find the exact values of the missing side lengths:
The “short leg” is half the hypotenuse
π / 3
60° 1
45°
π / 4
30°-60°-90° 1
π / 6
30°
45°-45°-90° OR Isosceles Right
The “long leg” is the short leg multiplied by √3
45°
π / 4
The hypotenuse is any leg multiplied by √2

3. Exact Coordinates on the Unit Circle
The angles that have the same reference angles as the angles from special right triangles have exact coordinates
The angles from the special right triangles have exact coordinates 1
π / 2
90°
120°
2π / 3
π / 3
60°
135°
3π / 4
π / 4
45°
150°
5π / 6
π / 6
30° π
180° 0° -1 1
The x and y-intercepts obviously have exact coordinates
210°
7π / 6
11π / 6
330°
225°
5π / 4
7π / 4
315°
4π / 3
5π / 3
240°
300°
3π / 2
270° -1

4. Exact Coordinates on the Unit Circle1
π / 2
2π / 3
π / 3
1/2 1
60°
3π / 4 1
45°
π / 4
These coordinates tell you the exact values of cosine and sine for 16 angles.
5π / 6
1/2 1
30°
π / 6
They need to be memorized. π -1 1
7π / 6
11π / 6
5π / 4
7π / 4
4π / 3
5π / 3
3π / 2 -1

5. Exact Coordinates on the Unit Circle1
90°
120°
60°
135°
45°
These coordinates tell you the exact values of cosine and sine for 16 angles.
150°
30°
They need to be memorized.
180° 0° -1 1
210°
330°
225°
315°
240°
300°
270° -1

6. NOTE
The coordinates on that graph tell you the exact values of cosine and sine for 16 angles. They need to be memorized for all of the included angles. If you do not wish to memorize the unit circle or use special right triangles, the following is a trick to assist in memorization.

7. Reference Angle
On the left are 3 reference angles that we know exact trig values for. To find the reference angle for angles not in the 1st quadrant (the angles at right), ignore the integer in the numerator.
NOTE: Multiply the number in the numerator by the degree to find the angle’s quadrant.

8. Example
Find the reference angle and quadrant of the following:
Or 45º

9. Stewart’s Table: Finding Exact Values of Trig Functions
R.A.
Sin
Cos
Tan
Find the value of the Reference Angle.
Find the angles quadrant to figure out the sign (+/-).
Each time the square root number goes up by 1
Reverse the order of the values from sine

10. How to Remember which Trigonometric Function is Positive1
Just Sine
All S A
STUDENTS
ALL -1 1
TAKE
CALCULUS T C
Just Tangent
Just Cosine -1

11. Example 1 Find the exact value of the following: Thought process
The only thing required for a correct answer (unless the question says explain)

12. Example 2 Isolate the Trig Function Are there more answers?
Find the exact solutions to the equation below if 0 ≤ x ≤ 2π:
The answer must be in Radians
Isolate the Trig Function
Are there more answers?
Find the answer in degrees first 1
120°
Find the Reference Angle
Convert the answers to radians
60° -1 1
60°
Use the reference angle to find where Cosine is also negative
180°+60° =240° -1

13. Example 3 Find the exact value of the following: Thought process
The only thing required for a correct answer (unless the question says explain)