1. Using Fundamental Identities
Objectives:
Recognize and write the fundamental trigonometric identities
Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions

2. WHY???
Fundamental trigonometric identities can be used to simplify trigonometric expressions, such as for the coefficient of friction.

3. Fundamental Trigonometric Identities
Reciprocal Identities
Quotient Identities

4. Fundamental Trigonometric Identities
Pythagorean Identities
Even/Odd Identities

5. Fundamental Trigonometric Identities
Cofunction Identities

6. Example: If and Ө is in quadrant II, find each function value.
a) sec Ө
To find the value of this function, look for an identity that relates tangent and secant.
Tip: Use Pythagorean Identities.

7. Example: If and Ө is in quadrant II, find each function value. (Cont.)
b) sin Ө
c) cot (- Ө )
Tip: Use Quotient Identities.
Tip: Use Reciprocal and Negative-Angle Identities.

8. 2. Use the values
cos x > 0 and identities to find the values of all six trigonometric functions.
What quadrant will you use?
1st quadrant

12. Using Identities to Evaluate a Function
Use the given values to evaluate the remaining trigonometric functions
(You can also draw a right triangle)

13. Simplify an Expression
Simplify cot x cos x + sin x.
Click for answer.

14. Example: Simplify
1. Factor csc x out of the expression.

15. 2. Use Pythagorean identities to simplify the expression in the parentheses.

16. 3. Use Reciprocal identities to simplify the expression.

17. Simplifying a Trigonometric Expression

18. Factoring Trigonometric Expressions
Factor the same way you would factor any quadratic.
If it helps replace the “trig” word with x
Factor the same way you would factor

19. Make it an easier problem.
Let a = csc x
2a2 – 7a + 6
(2a – 3)(a – 2)
Now substitute csc x for a.

20. 1. Use Pythagorean identities to get one trigonometric function in the expression.

21. 2. Now factor.

22. Factoring Trigonometric Expressions

23. More Factoring

24. Adding Trigonometric Expressions (Common Denominator)

25. Adding Trigonometric Expressions

26. Rewriting a Trigonometric Expression so it is not in Fractional Form

27. Trigonometric Substitution