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Pre – Calculus
Unit 4
Section 5.1 Notes – 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
sinx = 1 – cosx
𝑥−3
=𝑥+3
Example 1:
3
a) If cos θ = , find sec θ
4
5
3
b) If sec x = and tan x = , find sinx
4
Pythagorean Identities
Recall from 4.3 that trigonometric functions can be defined on a unit circle as shown.
Notice that for any angle, sine and cosine are directed lengths of the legs of a right triangle
with hypotenuse 1.
Apply the Pythagorean Theorem to this right triangle to establish another basic trig identity.
sin2 θ + cos2 θ = 12
(-sin θ)2 + (-cos θ)2 = 12
4
-
Other forms of the Pythagorean Identities
Example 2:
a) If cot θ = 2 and cos θ < 0, find sin θ and cos θ.
b) Find the value of cscθ and cotθ if tanθ = −
4
3
and cosθ < 0
Simplfying
• Helpful Hints
– No fractions
– 1 trig function
– If you see addition or subtraction with a 1 and/or a (trig function)2 will most likely use P.T. Identities
– Usually don’t rewrite sin or cos as their reciprocals, but will write tan, csc, sec, and cot
– Factor out a GCF
– Common Denominators / Conjugate
Example 4: Use identities to simplify
a)
1
cos 𝑥
(1 − sin2 𝑥)
b) cscx – cosx cotx
Example 5: Use identities to simplify
a) 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+cos 𝑥
sin 𝑥
Example 7: Rewrite as an expression that does not involve fractions.
a)
c)
1+tan2 𝑥
csc2 𝑥
csc 𝑥
1−sin 𝑥
b)
sin2 𝑥
1+cos 𝑥
+
sin 𝑥
1+cos 𝑥
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 − cos2 𝑥)(csc 𝑥)
d) sin x
PRACTICE:
1. sin θ sec θ cot θ
2. cot x sec x sin x
3. tan x csc x cos x
4.
tan 𝜃 cot 𝜃
csc 𝜃