1. 1
   The angle $\theta$ satisfies $\frac{\pi}{2}<\theta<\pi$ and $\sin\theta=\frac{3}{5}$.

   (a)

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

   [1 mark]

   • A$-\frac{5}{4}$
   • B$\frac{5}{4}$
   • C$\frac{5}{3}$
   • D$-\frac{4}{5}$

   (b)

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

   [1 mark]

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

   (c)

   Find the exact value of $\operatorname{cosec}\theta+\cot\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   This question is about the inverse trigonometric functions $y=\arcsin x$, $y=\arccos x$ and $y=\arctan x$, each defined on its largest possible domain and taking its principal values.

   (a)

   Find the exact value of $\arcsin\left(-\frac{\sqrt{3}}{2}\right)+\arccos\left(-\frac{1}{2}\right)$.

   [1 mark]

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

   (b)

   Find the range of the function $y=2\arctan x+\frac{\pi}{2}$, $x\in\mathbb{R}$.

   [1 mark]

   • A$-\frac{\pi}{2}\le y\le\frac{3\pi}{2}$
   • B$0<y<\pi$
   • C$-\frac{\pi}{2}<y<\frac{3\pi}{2}$
   • D$-\frac{3\pi}{2}<y<\frac{\pi}{2}$

   (c)

   Find the largest possible domain of the function $y=\arccos(2x-3)$, and write down its range.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An observer stands on level ground $40$ m from the point directly below a drone. The drone rises vertically, and when it is $h$ metres above the ground its angle of elevation from the observer is $\theta$, where $0<\theta<\frac{\pi}{2}$, so that $\theta=\arctan\left(\frac{h}{40}\right)$. Ignore the height of the observer.

   (a)

   Find the exact value of $h$ when $\theta=\frac{\pi}{3}$, and explain why $\theta$ can never equal $\frac{\pi}{2}$ however high the drone rises.

   [3 marks]

   (b)

   Use the identity $1+\cot^{2}\theta=\operatorname{cosec}^{2}\theta$ to show that $\operatorname{cosec}\theta=\dfrac{\sqrt{h^{2}+1600}}{h}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(\theta)=\sec^{2}\theta-\tan\theta$, for $0\le\theta\le\pi$, $\theta\neq\frac{\pi}{2}$.

   (a)

   Solve the equation $f(\theta)=3$, giving your answers in exact form.

   [6 marks]

   (b)

   Find the minimum value of $f(\theta)$ and the exact value of $\theta$ at which it occurs. Hence explain why the equation $f(\theta)=\frac{1}{2}$ has no solutions.

   [6 marks]

   Total for question 4: 12 marks

End of questions