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

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

Trig Identities

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

Example 1: C) If cot x = 2 5 5 and sin x = 5 3 , find cos x.

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

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

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.

Cofunction Identities

Odd-Even Identities

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

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

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

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

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

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

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

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.

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.