1. Section 5.1 Trigonometric Identities
Objectives:
- Identify basic trig identities
Use basic trig identities to
find trig values
simplify and rewrite expressions

2. Identities
An equation is an identity if the left side is equal to the right side for all values of the variable for which both sides are defined.
IDENTITY NON-IDENTITY
𝑥 2 −9 𝑥−3 =𝑥+3 sinx = 1 – cosx

3. Trig Identities

4. Example 1:
A) If cos  = 3 4 , find sec  B) If sec x = 5 4 and tan x = 3 4 , find sinx.

5. Example 1:
C) If cot x = and sin x = , find cos x.

6. Pythagorean Identities
(sin )2 + (cos )2 = 12
sin2  + cos2  = 1

7. Pythagorean Identities
sin2  + cos2  = 1 tan2  + 1 = sec2  cot2  + 1 = csc2 

8. Example 2:
A) If cot θ = 2 and cos θ < 0, find sin θ and cos θ. B) Find the value of csc and cot if tan = − 4 3 and cos < 0.

9. Cofunction Identities

10. Odd-Even Identities

11. Example 3:
A) If cos x = –0.75, find B) If cos x = 0.73, find

12. Simplifying Helpful Hints No fractions 1 trig function
Factor out a GCF
Common Denominators / Conjugate

13. Example 4: Use identities to simplify
A) 1 cos 𝑥 (1− sin 2 𝑥)
B) cscx – cosx cotx

14. Example 5: Use identities to simplify
cos x tan x – sin x cos 2 x
B) cos x sin2x - cosx

15. Example 6: Use identities to simplify
A) sec 𝑥 1− sec 𝑥 − sec 𝑥 1+ sec 𝑥 B) 1+𝑐𝑜𝑠𝑥 sin 𝑥 + sin 𝑥 1+ cos 𝑥

16. Example 7: Rewrite as an expression that does not involve fractions
A) tan 2 𝑥 csc 2 𝑥
B) sin 2 𝑥 1+ cos 𝑥

17. Match the trigonometric identity with one of the expressions: 1
Match the trigonometric identity with one of the expressions: 1. sec x cos x a) sec x 2. tan x csc x b) –1 3. cot 2 𝑥− csc 2 𝑥 c) 1 4. (1− cos 2 𝑥)( csc 𝑥) d) sin x

18. PRACTICE: 1. sin θ sec θ cot θ 2. cot x sec x sin x 3
PRACTICE: 1. sin θ sec θ cot θ 2. cot x sec x sin x 3. tan x csc x cos x 4.

19. Put the steps in order
In a small group, you receive a bag of problems. Each bag contains 3 problems that need to be simplified using identities. Each simplified step is provided, however, you must put the steps in order for each of the 3 problems.