1. 1
   Let $A=\frac{\pi}{4}$ and $B=\frac{\pi}{6}$. Do not use a calculator in this question.

   (a)

   Find the exact value of $\sin(A+B)$.

   [1 mark]

   • A$\frac{\sqrt{6}-\sqrt{2}}{4}$
   • B$\frac{\sqrt{2}+1}{2}$
   • C$\frac{\sqrt{6}+\sqrt{2}}{4}$
   • D$\frac{\sqrt{6}+\sqrt{2}}{2}$

   (b)

   Find the exact value of $\tan(A-B)$.

   [1 mark]

   • A$2+\sqrt{3}$
   • B$1-\frac{\sqrt{3}}{3}$
   • C$\frac{\sqrt{6}-\sqrt{2}}{4}$
   • D$2-\sqrt{3}$

   (c)

   Find the exact value of $\cos(2A+B)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The angle $x$ satisfies $\pi<x<\frac{3\pi}{2}$ and $\tan x=\frac{3}{4}$.

   (a)

   Find the exact value of $\tan2x$.

   [1 mark]

   • A$\frac{24}{7}$
   • B$-\frac{24}{7}$
   • C$\frac{3}{2}$
   • D$\frac{7}{24}$

   (b)

   Find the exact value of $\sin2x$.

   [1 mark]

   • A$-\frac{24}{25}$
   • B$\frac{24}{25}$
   • C$\frac{12}{25}$
   • D$\frac{7}{25}$

   (c)

   Find the exact value of $\cos\left(x+\frac{\pi}{3}\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The compound angle identities for $\sin(A+B)$ and $\cos(A+B)$ are given in the formula booklet. In this question you should derive the results from them.

   (a)

   Use the identity for $\cos(A+B)$ to show that $\cos2\theta=1-2\sin^{2}\theta$.

   [3 marks]

   (b)

   Hence, by writing $3\theta=2\theta+\theta$, show that $\sin3\theta=3\sin\theta-4\sin^{3}\theta$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A painting hangs on a vertical wall. Its bottom edge is $1$ m above a visitor's eye level and its top edge is $3$ m above eye level. The visitor stands $x$ m from the wall, where $x>0$. The angles of elevation from the visitor's eye to the top and bottom edges are $\alpha$ and $\beta$ respectively, and the angle the painting subtends at the eye is $\theta=\alpha-\beta$.

   (a)

   Show that $\tan\theta=\dfrac{2x}{x^{2}+3}$, and hence find the value of $\theta$ when $x=\sqrt{3}$.

   [6 marks]

   (b)

   (i) Show that $x^{2}+3\ge2\sqrt{3}\,x$ for all real $x$.\n(ii) Hence show that $\theta\le\frac{\pi}{6}$ for all $x>0$, and state the distance from the wall at which the painting subtends the greatest angle.

   [6 marks]

   Total for question 4: 12 marks

End of questions