Target Mathematics

1.79: Trigonometric identities and areas using trigonometry

4PM1/1/January/2018 — Question 6 · 11 marks

1.79 diagram 1
Figure 1 shows the triangle ABCABC with AB=x cmAB=x\text{ cm}, BC=(2x−2) cmBC=(2x-2)\text{ cm}, AC=(x+4) cmAC=(x+4)\text{ cm} and ∠BAC=θ∘\angle BAC=\theta^\circ.
Given that tan⁡θ∘=255\tan\theta^\circ=\sqrt{255} and without finding the value of θ\theta,
(a) show that cos⁡θ∘=116\cos\theta^\circ=\dfrac1{16}. (2)
Hence find
(b) the value of xx, (5)
(c) the size, in degrees to 1 decimal place, of ∠ABC\angle ABC, (2)
(d) the area, in cm2\text{cm}^2 to 3 significant figures, of triangle ABCABC. (2)