1. 1
   The angle $\theta$ is obtuse, and $\sin\theta = \frac{5}{13}$.

   (a)

   Find the exact value of $\cos\theta$.

   [1 mark]

   • A$\frac{12}{13}$
   • B$-\frac{8}{13}$
   • C$\frac{8}{13}$
   • D$-\frac{12}{13}$

   (b)

   Find the exact value of $\sin 2\theta$.

   [1 mark]

   • A$\frac{120}{169}$
   • B$\frac{10}{13}$
   • C$-\frac{120}{169}$
   • D$-\frac{60}{169}$

   (c)

   Find the exact value of $\cos 2\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The angle $\alpha$ satisfies $\pi < \alpha < \frac{3\pi}{2}$ and $\cos\alpha = -\frac{2}{3}$.

   (a)

   Find the exact value of $\tan\alpha$.

   [1 mark]

   • A$-\frac{\sqrt{5}}{2}$
   • B$\frac{\sqrt{5}}{2}$
   • C$\frac{2}{\sqrt{5}}$
   • D$-\frac{2}{\sqrt{5}}$

   (b)

   Find the exact value of $\cos 2\alpha$.

   [1 mark]

   • A$-\frac{1}{9}$
   • B$\frac{1}{9}$
   • C$-\frac{4}{3}$
   • D$\frac{17}{9}$

   (c)

   Find the exact value of $\sin 2\alpha$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $f(x) = \frac{\sin 2x}{1 + \cos 2x}$, for $0 < x < \frac{\pi}{2}$.

   (a)

   Show that $f(x) = \tan x$.

   [3 marks]

   (b)

   Given that $\cos 2x = \frac{7}{25}$, find the exact value of $f(x)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A garden sprinkler sends a jet of water from ground level at an angle $\theta$ above the horizontal, where $0 < \theta < \frac{\pi}{2}$. The water lands a horizontal distance $R = 20\sin 2\theta$ metres from the sprinkler, and the jet reaches a maximum height of $H = 10\sin^2\theta$ metres.

   (a)

   The sprinkler is set so that $\tan\theta = \frac{1}{2}$.
   (i) Find the exact values of $\sin\theta$ and $\cos\theta$.
   (ii) Hence find the values of $R$ and $H$.

   [6 marks]

   (b)

   (i) Show that $\frac{H}{R} = \frac{1}{4}\tan\theta$.
   (ii) The sprinkler is reset so that the water lands at a horizontal distance equal to three times the maximum height of the jet. Find the exact value of $\cos 2\theta$, and state what the sign of your answer tells you about $\theta$.

   [6 marks]

   Total for question 4: 12 marks

End of questions