1. OBJECTIVES: Evaluate trigonometric functions of any angle using reference angles. Trigonometric Functions of Non-Acute Angles

2. Special Angles Certain special angles, such as the quadrantals, and, occur so often in trigonometry and in more advanced mathematics, that they deserve special study. Therefore, we find the EXACT TRIG function values of these angles. We don’t use the calculator, because it gives only approximate values. To help find the exact values, we use properties of Geometry and the Pythagorean Theorem.

3. Two special right triangles

4. Quadrantal Angles

5. Reference Angles The values of the trigonometric functions of angles greater than ( or less than ) can be determined from their values at corresponding acute angles called reference angles. Reference Angle:Let be an angle in standard position. Its reference angle is the positive acute angle formed by the terminal side of and the horizontal axis. If an angle is negative or has a measure greater than, its reference angle is found by first finding its coterminal angle that is between

6. Reference Angles

7. EX 1: Find the reference angles for the following angles. A)E) B)F) C) D)

8. Reference angles and Special angles By using reference angles and the special angles, you can greatly extend the scope of EXACT trigonometric values. For instance, knowing the function values of means that you know the function values of all angles for which is a reference angle. The values of the trig functions for any non-quadrantal angle can be determined by finding the function values for an angle between. Therefore, we have placed them on a UNIT CIRCLE to use as a guide sheet. Learning the UNIT CIRCLE is worth the effort because doing so will increase both your efficiency and your confidence.

9. Algorithm for Finding Exact Values 1.In which quadrant is the angle located? 2.Is the trigonometric function positive or negative in that quadrant? 3.What is the reference angle? 4.What is the exact value for the reference?

10. EX 2: Find the EXACT values of the trig functions for the following angles A) B)

11. EX 2: Find the EXACT values of the trig functions for the following angles C) D)

12. EX 3: Find two solutions of equation. Give your answers in degrees and radians A) B)