1. Trigonometric Identities M 120 Precalculus V. J. Motto

2. Preliminary Comments Remember an identity is an equation that is true for all defined values of a variable We are going to use the identities that we have already established to "prove" or establish other identities. Let's summarize the basic identities we have.

3. Right Triangle Definitions

4. Unit Circle Definitions

5. Basic Trigonometric Identities Reciprocal Identities

6. Basic Trigonometric Identities Quotient Identities

7. Basic Trigonometric Identities Pythagorean Identities

8. Basic Trigonometric Identities Even-Odd Identities

9. Establish the following identity: In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match. Let's sub in here using reciprocal identity We often use the Pythagorean Identities solved for either sin 2  or cos 2 . sin 2  + cos 2  = 1 solved for sin 2  is 1 - cos 2  which is our left-hand side so we can substitute. We are done! We've shown the LHS equals the RHS

10. Establish the following identity: Let's sub in here using reciprocal identity and quotient identity Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom We worked on LHS and then RHS but never moved things across the = sign combine fractions FOIL denominator

11. Find common denominators when there are fractions. Squared functions often suggest Pythagorean Identities. Work on the more complex side first. A denominator of 1 + trig function suggest multiplying top & bottom by conjugate which leads to the use of Pythagorean Identity. When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities. Trigonometric Identities are like puzzles! They are fun and test you algebra skills and insights. Enjoy them! Attitude does make a difference in success. Hints for Establishing Identities

12. Other Trigonometric Identities Identities expressing trigonometric function in terms of their complements.

13. Other Trigonometric Identities Sum formulas of sine and cosine The derivation involves the use of geometry.

14. Other Trigonometric Identities Double angle formulas for sine and cosine These are easily derived from the previous identities.