1. TRIGONOMETRIC IDENTITIES

2. TRIGONOMETRIC IDENTITIES DEFINITION: A trigonometric identity is an equation involving the trigonometric functions that holds for all values of the variable. BASIC TRIGONOMETRIC IDENTITIES Reciprocal Identities Pythagorean Identities

3. Quotient or Ratio Identities Negative Argument Identities

4. There is no set procedure to prove identities. However, there are several strategies to use when proving identities. 1. Know the fundamental identities and look for ways to apply them. Write all expressions in terms of sine and cosine. If you choose to work with only one side of an identity, continuously refer back to the other side to see what you are trying to obtain. When one side contains only one trigonometric function, attempt to rewrite all the functions on the other side in terms of that function. It is usually easier to start with the more complicated side. Use Pythagorean identities to substitute for the expression equal to 1.

5. 6. Perform algebraic operations. a) Factoring. b) Simplifying complex rational expressions. c) Finding the LCD and combining fractions. d) Combining like terms. e) Multiplying both the numerator and denominator by the same expression to obtain an equivalent fraction. f) Replacing a binomial with a monomial. Note: Proving an identity is not the same as solving an equation. This means you can’t perform operations such as adding the same expression to both sides or dividing both sides by the same expression. These operations apply only to an equation where the statement is known to be true; an identity must be proven to be true.

6. EXAMPLE: Prove the following identities.

7. Sum and Difference Identities

8. EXAMPLE: Use sum or difference identities to find theexact value of the given function. II. Given that , , , and , find the following:

9. III. Write each expression in terms of a trigonometric function of one angle. IV. Prove each identity.

10. Double - Angle Identities EXAMPLE: I. Write each expression in terms of a trigonometric function of one angle.

11. II. Use double-angle identities to find theexactvalue of the given function. III. Prove each identity.

12. Half - Angle Identities EXAMPLE: I. Use half-angle identities to find the exact values of the following.

13. II. Find the exact value of each trigonometric function. Assume .

14. Product / Sum Identities

15. EXAMPLE: I. Express each product as a sum or difference. II. Express each sum or difference as a product. III. Express each sum or difference as a product.