1.65: Show can be that written the equation in the form a sin(x sin(x a a

4PM1/1/January/2019 — Question 4 · 11 marks

1.65 diagram 1
sin⁡(A+B)=sin⁡Acos⁡B+sin⁡Bcos⁡A\sin(A+B)=\sin A\cos B+\sin B\cos A
tan⁡A=sin⁡Acos⁡A\tan A=\frac{\sin A}{\cos A}
(a) Show that the equation
asin⁡(x−30)∘=bsin⁡(x+30)∘a\sin(x-30)^\circ=b\sin(x+30)^\circ
can be written in the form
tan⁡x∘=a+b3(a−b).\tan x^\circ=\frac{a+b}{\sqrt{3}(a-b)}.
(5)
In triangle ABC\displaystyle ABC, AC=6 cm\displaystyle AC=6\,\mathrm{cm}, BC=14 cm\displaystyle BC=14\,\mathrm{cm}, ∠ABC=(x−30)∘\displaystyle \angle ABC=(x-30)^\circ and ∠BAC=(x+30)∘\displaystyle \angle BAC=(x+30)^\circ, as shown in Figure 2.
(b) Find, in degrees to 1 decimal place, the size of ∠ACB\displaystyle \angle ACB.
(4)
(c) Find, to 3 significant figures, the area of triangle ABC\displaystyle ABC.
(2)