1.57: Trigonometric Equations and Identities

4PM1/1/June/2019 — Question 7 · 12 marks

(a) Solve
(3cos⁡θ+5)(5sin⁡θ−3)=0(3\cos\theta+5)(5\sin\theta-3)=0
for 0∘⩽θ<180∘\displaystyle 0^\circ\leqslant\theta<180^\circ.
Give your answers in degrees to 1 decimal place.
(2)
(b) Show that
8sin⁡(x−α)=3sin⁡(x+α)8\sin(x-\alpha)=3\sin(x+\alpha)
can be written in the form
5tan⁡x=11tan⁡α.5\tan x=11\tan\alpha.
(5)
(c) Hence solve
8sin⁡(2y−30∘)=3sin⁡(2y+30∘)8\sin(2y-30^\circ)=3\sin(2y+30^\circ)
for 0∘⩽y<180∘\displaystyle 0^\circ\leqslant y<180^\circ.
Give your answers in degrees to 1 decimal place.
(5)