Target Mathematics

1.80: Trigonometric identities, trigonometric equations and exact trigonometric values

4PM1/1/January/2018 — Question 10 · 16 marks

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)\cos^2\theta=\dfrac12(\cos2\theta+1). (3)
Given that f(θ)=8cos⁡4θ+8sin⁡2θ−7f(\theta)=8\cos^4\theta+8\sin^2\theta-7
(b) show that f(θ)=cos⁡4θf(\theta)=\cos4\theta. (5)
(c) Solve, for 0≤θ≤π20\le\theta\le\dfrac\pi2, the equation
16cos⁡4(θ−π6)+16sin⁡2(θ−π6)−15=016\cos^4\left(\theta-\dfrac\pi6\right)+16\sin^2\left(\theta-\dfrac\pi6\right)-15=0
(4)
(d) Using calculus, find the exact value of
∫0π/2(8cos⁡4θ+8sin⁡2θ+2sin⁡2θ) dθ\displaystyle\int_0^{\pi/2}(8\cos^4\theta+8\sin^2\theta+2\sin2\theta)\,d\theta
(4)