1. 5.1/5.2 – Using Fundamental Identities and Verifying Trigonometric Identities

2. In this section, you will learn to:
Evaluate trigonometric functions
Simplify trigonometric expressions
Develop additional trigonometric identities
Solve trigonometric equations

3. Fundamental Trigonometric Identities:
a) Reciprocal Functions:
b) Quotient Identities:

4. Fundamental Trigonometric Identities:
c) Pythagorean Identities:

5. Fundamental Trigonometric Identities:
d) Co-function Identities:

6. Even Functions:

7. Even Functions:

8. Odd Functions:

9. Odd Functions:

10. Odd Functions:

11. Odd Functions:

12. Fundamental Trigonometric Identities:
e) Even Functions:
f) Odd Functions:

13. Guidelines for verifying trigonometric identities:
Work with one side of the equation at a time. It is often better to work with the more complicated side first.
Look for expressions that can be factored, adding fractions, multiplying binomials, create a new denominator or multiply by the conjugate.

14. Guidelines for verifying trigonometric identities:
Look for ways to substitute reciprocal functions and fundamental identities.
Use the final expressions to give you hints to which expressions need to be converted / substituted to arrive at the final answer.
When all else fails, work on both sides of the equation.

15. Example #1:  
Use the given values to evaluate (if possible) the remaining trigonometric functions. Use reciprocal functions/fundamental identities to solve.

16. Example #1:  
Because sine is negative and secant is positive, the angle must lie in quadrant IV. In quadrant IV, cosine and secant are positive and tangent / cotangent / sine /cosecant are negative.

17. Example #1:  

18. Example #2:
Use addition or subtraction to substitute a fundamental identity to simplify the expression without a fractional expression.

19. Example #2:

20. Example #3:
Use factoring to substitute a fundamental identity to simplify the expression without a fractional expression.

21. Example #3:

22. Example #4: Verify the trigonometric identity.

23. Example #4: Verify the trigonometric identity.

24. Example #4: Verify the trigonometric identity.

25. Example #4: Verify the trigonometric identity.

26. Example #4: Verify the trigonometric identity.

27. Example #4: Verify the trigonometric identity.

28. Example #4: Verify the trigonometric identity.