• A Level Maths

Demo

• A Level
• Trigonometry (Inverse)

1. 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. $\dfrac{\pi}{2}$

2. 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. $2$

3. 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 $\alpha$.

$\alpha, \pi - \alpha$

4. 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. $\dfrac{x}{\sqrt{1-x^2}}$

5. Sketch:

1. $y=\dfrac{\pi}{2}+2\arcsin x$
2. $y = \pi - \arctan x$

6. Sketch:

1. $y=\arccos(2x+1)$
2. $y=-2\arcsin(-x)$

7. Sketch $$y=\mathrm{arcsec}\ x$$ and state its range.

$0 \leqslant y \leqslant \pi$, $y \neq \frac{\pi}{2}$

8. Evaluate: $$\displaystyle\int_0^{\frac{\pi}{2}}\sin x\ \mathrm{d}x + \int_0^1\arcsin x\ \mathrm{d}x$$

$\dfrac{\pi}{2}$

9. Sketch $$y = 3\arcsin x - \dfrac{\pi}{2}$$ showing exactly the end points of the curve and any points where it crosses the axes.

10. Simplify $$\tan(\arctan 3 - \arctan 2)$$

$\dfrac{1}{7}$

11. Given that, for $-1 \leqslant x < 0$, $$\arccos x = \arctan\dfrac{\sqrt{1-x^2}}{x} + k$$ find $k$.

$\pi$

12. Find exact solutions to:

1. $\arcsin x = \arccos 2x$
2. $2\arccos 2x = \arccos x$

1. $\sqrt{\dfrac{1}{5}}$
2. $\dfrac{1+\sqrt{33}}{16}$