Target Mathematics

1.104: Trigonometric identities and equations

4PM1/1/June/2016 — Question 5 · 9 marks

Using the identities
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)=\sin A\cos B+\cos A\sin B
tan⁡A=sin⁡Acos⁡A\tan A=\dfrac{\sin A}{\cos A}
(a) show that the equation
3sin⁡(x+α)=5sin⁡(x−α)3\sin(x+\alpha)=5\sin(x-\alpha)
can be written in the form tan⁡x=4tan⁡α\tan x=4\tan\alpha. (5)
(b) Hence solve, to the nearest integer, the equation
3sin⁡(2y+30)∘=5sin⁡(2y−30)∘for 90≤y<1803\sin(2y+30)^\circ=5\sin(2y-30)^\circ\quad\text{for }90\le y<180
(4)