Q1
Answer
Find the exact value of:
  1. $\arcsin\left(\dfrac{1}{2}\right) + \arcsin\left(-\dfrac{1}{2}\right)$
  2. $\arccos\left(\dfrac{1}{2}\right) + \arctan\left(\dfrac{1}{\sqrt{3}}\right)$
  1. $0$
  2. $\dfrac{\pi}{2}$
Q2
Answer
Find the exact value of:
  1. $\cos(\arcsin(-0.5))$
  2. $\sec\left(\arctan\left(\sqrt{3}\right)\right)$
  1. $\dfrac{\sqrt{3}}{2}$
  2. $2$
Q3
Answer
Given that $$\arcsin k = \alpha, \quad 0< k < 1$$ and $\alpha$ is in radians, find the first two positive values of $x$ for which $\sin x = k$ in terms of $\alpha$.
$\alpha, \pi - \alpha$
Q4
Answer
Given $$\mathrm{arcsin}\ x = k \quad 0 < k < \frac{\pi}{2}$$
  1. Write $\cos k$ in terms of $x$.
  2. Write $\tan k$ in terms of $x$.
  1. $\sqrt{1-x^2}$
  2. $\dfrac{x}{\sqrt{1-x^2}}$
Q5
Answer
Sketch:
  1. $y=\dfrac{\pi}{2}+2\arcsin x$
  2. $y = \pi - \arctan x$

Q6
Answer
Sketch:
  1. $y=\arccos(2x+1)$
  2. $y=-2\arcsin(-x)$

Q7
Answer
Sketch $$y=\mathrm{arcsec}\ x$$ and state its range.
$0 \leqslant y \leqslant \pi$, $y \neq \frac{\pi}{2}$
Q8
Answer
Evaluate: $$\displaystyle\int_0^{\frac{\pi}{2}}\sin x\ \mathrm{d}x + \int_0^1\arcsin x\ \mathrm{d}x$$
$\dfrac{\pi}{2}$
Q9
Answer
Sketch $$y = 3\arcsin x - \dfrac{\pi}{2}$$ showing exactly the end points of the curve and any points where it crosses the axes.

Q10
Answer
Simplify $$\tan(\arctan 3 - \arctan 2)$$
$\dfrac{1}{7}$
Q11
Answer
Given that, for $-1 \leqslant x < 0$, $$\arccos x = \arctan\dfrac{\sqrt{1-x^2}}{x} + k$$ find $k$.
$\pi$
Q12
Answer
Find exact solutions to:
  1. $\arcsin x = \arccos 2x$
  2. $2\arccos 2x = \arccos x$
  1. $\sqrt{\dfrac{1}{5}}$
  2. $\dfrac{1+\sqrt{33}}{16}$