Target Mathematics

1.87: Trigonometric identities, trigonometric equations and exact trigonometric values

4PM1/2/June/2018 — Question 8 · 16 marks

cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B)=\cos A\cos B-\sin A\sin B
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)=\sin A\cos B+\cos A\sin B
Using the above identities
(a) show that
(i) cos⁡2θ=1−2sin⁡2θ\cos2\theta=1-2\sin^2\theta
(ii) sin⁡2θ=2sin⁡θcos⁡θ\sin2\theta=2\sin\theta\cos\theta (3)
f(θ)=cos⁡4θ+2cos⁡2θf(\theta)=\cos4\theta+2\cos2\theta
(b) Show that f(θ)=8sin⁡4θ−12sin⁡2θ+3f(\theta)=8\sin^4\theta-12\sin^2\theta+3. (4)
(c) Solve, giving your solutions to 3 significant figures, the equation
4sin⁡4x∘−6sin⁡2x∘−cos⁡2x∘+1.2=0,0≤x<904\sin^4x^\circ-6\sin^2x^\circ-\cos2x^\circ+1.2=0,\qquad 0\le x<90
(4)
(d) (i) Find ∫(2sin⁡4θ−3sin⁡2θ) dθ\displaystyle\int(2\sin^4\theta-3\sin^2\theta)\,d\theta
(ii) Hence find the exact value of ∫0π/3(2sin⁡4θ−3sin⁡2θ) dθ\displaystyle\int_0^{\pi/3}(2\sin^4\theta-3\sin^2\theta)\,d\theta.
Give your answer in the form ab−cπa\sqrt b-c\pi where aa and cc are rational numbers and bb is a prime number. (5)