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. Fundamental Identities

3. Fundamental Trigonometric Identities
Pythagorean Identities

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

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

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

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

8. Simplifying a Trigonometric Expression

9. 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

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

11. Factoring Trigonometric Expressions

12. Adding Trigonometric Expressions (Common Denominator)

13. Fundamental Identities

14. Simplifying Trigonometric Expressions
Claim:
Proof:

15. 4.2 Verifying Trigonometric Identities
Verifying identities
Testing identities using a graphing calculator

16. Verifying Identities
Verify right-to-left:

17. Verifying Identities Using a Calculator
Graph both sides of the equation in the same viewing window. If they produce different graphs they are not identities. If they appear the same the identity must still be verified.
Example:

18. Basic Trigonometric Identities
In this powerpoint, we will use trig identities to verify and prove equations

28. c) tan x sin x + cos x = sec x
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x
= sec x + 1
1 + sec x
L.S. = R.S.
b) sec x = tan x csc x
c) tan x sin x + cos x = sec x
L.S. = R.S.
L.S. = R.S.
5.4.8

29. = (sin2x - cos2x)(sin2x + cos2x) = (1 - cos2x - cos2x) = 1 - 2cos2x
Proving an Identity
d) sin4x - cos4x = cos2 x
= (sin2x - cos2x)(sin2x + cos2x)
= (1 - cos2x - cos2x)
= 1 - 2cos2x
1 - 2cos2x
L.S. = R.S. e)
L.S. = R.S.
5.4.9

30. Proving an Identity f)
L.S. = R.S.
5.4.10