A-LevelMathematicsTrigonometry9709/23 · Oct/Nov 2021 · Q79 Marks
By first expanding cos(2θ + θ), show that cos 3θ = 4 cos³ θ – 3 cos θ.
- (a)By first expanding cos(2θ + θ), show that cos 3θ = 4 cos³ θ – 3 cos θ.[3]
- (b)Find the exact value of 2 cos³ (π/18) – 3/2 cos(π/18).[2]
- (c)Find ∫ (12 cos³ x – 4 cos³ 3x) dx.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.
- [B1] (a) State cos 2θ cos θ – sin 2θ sin θ
- [M1] (a) Attempt correct relevant identities to express in terms of cos θ only (M0 if moving terms from side to side.)
- [A1] (a) Confirm 4 cos³ θ – 3 cos θ with sufficient detail (AG)
- [M1] (b) Use identity with θ = π/18
- [A1] (b) Obtain ½ cos (π/6) and hence √3/4
- [M1] (c) Express integrand in form k₁ (cos 3x + 3 cos x) + k₂ (cos 9x + 3 cos 3x)
- [A1] (c) Obtain correct integrand 9 cos x - cos 9x (OE (allow unsimplified).)
- [M1] (c) Integrate to obtain form k₃ sin x + k₄ sin 9x
- [A1] (c) Obtain correct 9 sin x – ⅓ sin 9x (Now simplified; condone missing ... + c .)
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