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

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

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

Exact Coordinates on the Unit Circle 1 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

Exact Coordinates on the Unit Circle 1 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

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.

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.

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

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

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

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

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

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