Target Mathematics

1.76: Trigonometric identities, trigonometric equations and exact trigonometric values

4PM1/1/June/2017 — Question 9 · 15 marks

Using cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B)=\cos A\cos B-\sin A\sin B,
(a) show that cos⁡2θ=12(cos⁡2θ+1)\displaystyle\cos^2\theta=\frac12(\cos2\theta+1). (2)
f(θ)=8cos⁡4θ+4cos⁡2θ−5f(\theta)=8\cos^4\theta+4\cos2\theta-5
(b) show that f(θ)=cos⁡4θ+6cos⁡2θf(\theta)=\cos4\theta+6\cos2\theta. (4)
Hence
(c) solve, for 0∘≤x<180∘0^\circ\le x<180^\circ, the equation
8cos⁡4x+4cos⁡2x−6cos⁡2x=4.58\cos^4x+4\cos^2x-6\cos2x=4.5
(4)
(d) find
(i) ∫f(θ) dθ\displaystyle\int f(\theta)\,d\theta
(ii) the exact value of ∫0π/3f(θ) dθ\displaystyle\int_0^{\pi/3}f(\theta)\,d\theta. (5)