Target Mathematics

1.106: Trigonometric identities, trigonometric equations and exact trigonometric values

4PM1/2/June/2016 — Question 9 · 16 marks

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)=\sin A\cos B+\cos A\sin B
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B)=\cos A\cos B-\sin A\sin B
Using the above identities
(a) show that cos⁡2θ=2cos⁡2θ−1\cos2\theta=2\cos^2\theta-1. (3)
(b) find a simplified expression for sin⁡2θ\sin2\theta in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta. (1)
(c) show that cos⁡3θ=4cos⁡3θ−3cos⁡θ\cos3\theta=4\cos^3\theta-3\cos\theta. (4)
Hence, or otherwise,
(d) solve, for 0≤θ<π0\le\theta<\pi, giving your answers in terms of π\pi, the equation
6cos⁡θ−8cos⁡3θ+1=06\cos\theta-8\cos^3\theta+1=0
(4)
(e) find
(i) ∫(8cos⁡3θ+4sin⁡θ) dθ\displaystyle\int(8\cos^3\theta+4\sin\theta)\,d\theta
(ii) the exact value of ∫0π/3(8cos⁡3θ+4sin⁡θ) dθ\displaystyle\int_0^{\pi/3}(8\cos^3\theta+4\sin\theta)\,d\theta (4)