Math 150 – Fall 2015
Section 8C
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Section 8D – Trigonometric Identities
Negative Angle Identities (cosine and secant are even functions and the others are
odd)
sin(−θ) = − sin θ
cos(−θ) = cos θ
tan(−θ) = − tan(θ).
csc(−θ) = − csc θ
sec(−θ) = sec θ
cot(−θ) = − cot(θ)
Complementary Angle Identities
sin
π
2
− θ = cos θ
tan
π
2
− θ = cot θ
sec
π
2
− θ = csc θ
cos
π
2
− θ = sin θ
cot
π
2
− θ = tan θ
csc
π
2
− θ = sec θ
Definition. The notation sin2 x is defined as
2
sin2 x = (sin x) = (sin x)(sin x)
We use this notation to prevent confusion with sin x2 = sin x2 .
Pythagorean Identities
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
1 + cot2 θ = csc2 θ
Sum of Two Angles Formulas
sin (x + y) = sin x cos y + cos x sin y
cos (x + y) = cos x cos y − sin x sin y
tan (x + y) =
tan x+tan y
1−tan x tan y
The difference of two angles formulas follow easily from the sum of two angles
formulas and negative angle identities.
Double Angle Formulas
sin 2α = 2 sin α cos α
cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α
tan 2α =
2 tan α
1−tan2 α
Math 150 – Fall 2015
Section 8C
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Additional Trigonometric Identities: There are many additional trigonometric
identities such as square formulas, half-angle formulas, product-to-sum formulas, and
sum-to-product formulas.
Proving Trigonometric Identities
Guidelines for Proving Trig Identities:
1. Start with one side, preferably the more complicated side (do NOT work both
sides at the same time!)
2. Use algebra and trigonometric identities to simplify the expressions.
3. If needed, try rewriting all expressions in terms of sine and cosine.
Example 1. Prove the following identities.
(a) tan2 θ + 1 = sec2 θ.
(b)
1−sin2 x
cos x
= cos x
(c) tan x sin x =
csc x
cot x+cot3 x
(d) cos 2x = cos2 x − sin2 x
Math 150 – Fall 2015
Section 8C
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Example 2. Evaluate the following.
(a) cos 7π
12
π
2π
π
(b) sin 2π
3 cos 12 + cos 3 sin 12
Example 3. Given cot θ =
−3
7
and cos θ < 0, exactly evaluate sin 2θ.
Example 4. Evaluate and fully simplify cos θ +
Example 5. Simplify sin2
−x
2
− cos2
−x
5
7π
4
if csc θ =
−6
5
and tan θ > 0.