Target Mathematics

1.91: Trigonometric identities and exact trigonometric values

4PM1/2/January/2015 — Question 4 · 7 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
(a) Write down the exact value of sin⁡45∘\sin45^\circ. (1)
Given that sin⁡θ=522\sin\theta=\dfrac{\sqrt5}{2\sqrt2} and cos⁡θ=322\cos\theta=\dfrac{\sqrt3}{2\sqrt2},
(b) show that sin⁡(45∘+θ)=3+54\sin(45^\circ+\theta)=\dfrac{\sqrt3+\sqrt5}{4}. (2)
(c) Find the exact value of cos⁡(45∘+θ)\cos(45^\circ+\theta). (2)
(d) Show that sin⁡(45∘+θ)×cos⁡(45∘+θ)=−18\sin(45^\circ+\theta)\times\cos(45^\circ+\theta)=-\dfrac18. (2)