Compound & Double Angle Identities (Grade 12) — Worksheet & Memo

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Grade 12 · Trigonometry

Compound & Double Angle Identities (Grade 12)

MEMORANDUM

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Date:

Mark

/ 26

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Expand cos(A − B).
(1)
cos A cos B + sin A sin B✓ (1)
2
Expand sin(A + B).
(1)
sin A cos B + cos A sin B✓ (1)
3
Determine the exact value of cos 75° without a calculator.
(3)
cos(45°+30°) = (√6 − √2)/4✓✓✓ (3)
4
Determine the exact value of sin 15° without a calculator.
(3)
sin(45°−30°) = (√6 − √2)/4✓✓✓ (3)
5
Write down the three forms of cos 2A.
(3)
cos²A − sin²A, 2 cos²A − 1, 1 − 2 sin²A✓✓✓ (3)
6
If sin θ = 3/5 and θ is acute, determine cos 2θ.
(3)
1 − 2(9/25) = 7/25✓✓✓ (3)
7
If sin θ = 3/5 and θ is acute, determine sin 2θ.
(3)
cos θ = 4/5, so 2(3/5)(4/5) = 24/25✓✓✓ (3)
8
Prove that (sin 2A)/(1 + cos 2A) = tan A, and state the restriction on A.
(5)
(2 sin A cos A)/(1 + (2 cos²A − 1)) ✓ = (2 sin A cos A)/(2 cos²A) ✓ = (sin A)/(cos A) ✓ = tan A ✓
The original denominator 1 + cos 2A = 2 cos²A is zero when cos A = 0, so cos A ≠ 0, i.e. A ≠ 90° + k·180°, k ∈ ℤ ✓(5)
9
Simplify 2 sin 15° cos 15°.
(2)
= sin 30° = 1/2✓✓ (2)
10
Why is sin(A + B) ≠ sin A + sin B?
(2)
Test A = 30°, B = 60°: sin 90° = 1 but sin 30° + sin 60° ≈ 1,37, so they are not equal.✓✓ (2)

TOTAL: 26 marks