Mathematics · Trigonometric identities and equations · 6 marks
f(θ) = 5 sin θ − 12 cos θ. (a) Express f(θ) in the form R sin(θ − α), where R > 0 and 0 < α < 90°. [3] (b) Hence state the maximum value of f(θ) and the smallest positive θ at which it occurs. [3]
Mark scheme
- (a) R = √(5² + 12²) = 13
- (a) tan α = 12/5
- (a) α = 67.4°, so f(θ) = 13 sin(θ − 67.4°)
- (b) Maximum value is 13
- (b) Occurs when sin(θ − 67.4°) = 1, so θ − 67.4° = 90°
- (b) θ = 157.4°
R is always positive and comes from Pythagoras. The maximum of R sin(…) is R, which makes part (b) one line once part (a) is right.