Finding the Exact Value of Trigonometric Functions

Review: Special Right Triangles 1 60° 30° 1 45 ° π / 6 π / 3 π / 4 Find the missing side Lengths:

0 3π / 2 π / 2 Important Points on Unit Circle ° 45° 60° 150° 135° 120° 210° 225° 240° 330° 315° 300° 0°0° 180° 90° π / 6 270° π / 4 π / 3 2π / 3 3π / 4 5π / 6 π 7π / 6 5π / 4 4π / 3 5π / 3 7π / 4 11π / 6

1 45° 1/2 1 30° 1/2 1 60° Important Points on Unit Circle 1 1 π / 6 π / 4 π / 3 π / 2 2π / 3 3π / 4 5π / 6 7π / 6 5π / 4 4π / 3 3π / 2 5π / 3 7π / 4 11π / 6 0 π

Important Points on Unit Circle ° 45° 60° 150° 135° 120° 210° 225° 240° 330° 315° 300° 0°0° 180° 90° 270°

Important Points on Unit Circle 1 1 π / 6 π / 4 π / 3 π / 2 2π / 3 3π / 4 5π / 6 7π / 6 5π / 4 4π / 3 3π / 2 5π / 3 7π / 4 11π / 6 0 π

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 1 st quadrant (the angles at right), ignore the integer in the numerator. Then multiply the number in the numerator by the degree to find the angle’s quadrant.

Stewart’s Table: Finding Exact Values of Trig Functions R.A.SinCosTan 1. Find the value of the Reference Angle. 2. 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

Example 1 Find the exact value of the following:

Example 2 Solve: π / 6

Slope on the Unit Circle 1 1 Ө (cosӨ,sinӨ) cosӨ sinӨ Slope = Opposite Adjacent What is the slope of the terminal side of an angle on the unit circle?

A Definition of Tangent There are values for which the tangent function are undefined: For any integer n. The tangent function is defined as: Any Θ that makes cos(Θ)=0. In general:

Stewart’s Table: Finding Exact Values of Trig Functions R.A.SinCosTan 1.Find the value of the Reference Angle. 2.Find the angles quadrant to figure out +/-. Each time the square root number goes up by 1 Reverse the order of the values from sine

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