Question 1

Paper 2, calculatorHard6 marks

The function ff is defined by

f(θ)=asin⁡θ+bcos⁡θ,0≤θ≤π,f(\theta)=a\sin\theta+b\cos\theta,\qquad 0\leq\theta\leq\pi,

where a,b∈Ra,b\in\mathbb R Figure 1 shows the graph of y=f(θ)y=f(\theta)

0π/6π/3π/22π/35π/6π-1.5-1-0.500.511.522.5θ (radians)f(θ)PQ
(a)

Determine the values of aa and bb

[3]
(b)

For 0<θ<π0<\theta<\pi θ≠π2\theta\ne\frac{\pi}{2} define

H(θ)=f(θ)f(π−θ)+tan⁡θ+tan⁡(π−θ).H(\theta)=f(\theta)f(\pi-\theta)+\tan\theta+\tan(\pi-\theta).

Determine whether HH attains a maximum. If it does not, state its supremum.

[3]

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