Pre Cal, Unit 4, Lesson03_student, page 1
Cosine composite angle Identities
Without proof we submit the identities for the cosine of the sum and
difference of angles:
cos(A – B) = cosA cosB + sinA sin B
cos(A + B) = cosA cosB – sinA sin B
These are often written compactly together as:
Example 1: Find cos(A + B) where sinA = –3/5, π ≤ A ≤ 3π/2; cosB = 1/5,
3π/2 ≤ B ≤ 2π.
The identity for the cosine of twice an angle, cos 2A= cos2A – sin2A, is
easily proved:
Making substitutions from sin2A + cos2A = 1, there are three forms of
the cosine of 2A:
cos 2A = cos2A – sin2A
cos 2A = 1 – 2sin2A
cos 2A = 2cos2A – 1
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Pre Cal, Unit 4, Lesson03_student, page 2
Example 2: Find cos2θ when sinθ = –15/17, π ≤ θ ≤ 3π/2.
The identity for the cosine of a half-angle is easily derived from
cos2θ = 2cos2θ –1:
Example 3: Find cos(A/2) when sinA = 4/5, π/2 ≤ A ≤ π.
Even and odd properties
cos(x) is an even function, so cos(–x) = cos(x) because all even
functions obey f(–x) = f(x).
sin(x) and tan(x) are odd functions, so sin(–x) = –sin(x) and
tan(–x) = –tan(x) because all odd functions obey f(–x) = –f(x).
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Pre Cal, Unit 4, Lesson03_student, page 3
Assignment:
1. Find cos(A – B) where sinA = –4/5, π ≤ A ≤ 3π/2; cosB = 3/4, 3π/2 ≤ B ≤ 2π.
2. Find cos2θ when cosθ = 1/3, 0 ≤ θ ≤ π/2.
3. Find cos(A/2) when cosA = –4/5, π/2 ≤ A ≤ π.
4. Simplify –sin(–3x)
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5. Simplify –cos(–5φ)
Pre Cal, Unit 4, Lesson03_student, page 4
6. Use the exact function values of 30o, 45o, and/or 60o to find cos(105o).
7. Use the exact function values of 30o, 45o, and/or 60o to find cos(22o 30’).
8. Use the exact function values of 30o, 45o, and/or 60o to find cos(120o).
9. Consider evaluating cos(A + B). If A and B are unknown but the sines and
cosines of these angles are known, what could be used to evaluate this composite
function?
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Pre Cal, Unit 4, Lesson03_student, page 5
Prove the following identities:
10. cos3A cos5A – sin3A sin5A = cos8A
11. (1 – 2sin2φ)2 = 1 – sin22φ
12. cos3A cosA – sin3AsinA = 2cos22A –1
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