Reciprocal trigonometric functions

Mark scheme

1. Answer. $2$, $2$ and $1$

   • $\sec \dfrac{\pi}{3} = \dfrac{1}{\cos \pi/3} = 2$.
   • $\operatorname{cosec} \dfrac{\pi}{6} = 2$ and $\cot \dfrac{\pi}{4} = 1$.

2. Answer. $x = \dfrac{\pi}{3}$ or $x = \dfrac{5\pi}{3}$

   • $\cos x = \dfrac{1}{2}$.
   • The solutions in the interval are $\dfrac{\pi}{3}$ and $\dfrac{5\pi}{3}$.

3. Answer. $\sec^2 x = 1 + \tan^2 x$ and $\operatorname{cosec}^2 x = 1 + \cot^2 x$.

   • Divide $\sin^2 x + \cos^2 x = 1$ by $\cos^2 x$ to obtain the secant identity.
   • Divide the same identity by $\sin^2 x$ to obtain the cosecant identity.

4. Answer. $x = \dfrac{\pi}{8}$ or $x = \dfrac{5\pi}{8}$

   • $\tan 2x = 1$, and $2x$ lies in $(0, 2\pi)$.
   • $2x = \dfrac{\pi}{4}$ or $\dfrac{5\pi}{4}$.

5. Answer. $x = 30^\circ$, $150^\circ$, $210^\circ$ or $330^\circ$

   • $\sin^2 x = \dfrac{1}{4}$, so $\sin x = \pm \dfrac{1}{2}$.
   • The four solutions in the interval are $30^\circ$, $150^\circ$, $210^\circ$ and $330^\circ$.

6. Answer. $x = 60^\circ$, $120^\circ$, $240^\circ$ or $300^\circ$

   • $\tan x = \sqrt{3}$ or $\tan x = -\sqrt{3}$.
   • The solutions are $60^\circ$, $180^\circ - 60^\circ$, $180^\circ + 60^\circ$ and $360^\circ - 60^\circ$.

7. Answer. $1$

   • $\cos x = -\dfrac{1}{\sqrt{2}}$, so $x = \dfrac{5\pi}{4}$ in the given interval.
   • At $\dfrac{5\pi}{4}$, $\tan x = 1$, so $\cot x = 1$.

8. Answer. $x = \arctan 2$, $\pi + \arctan 2$, $\pi - \arctan \dfrac{1}{2}$ or $2\pi - \arctan \dfrac{1}{2}$

   • Use $\sec^2 x = 1 + \tan^2 x$.
   • $2\tan^2 x - 3\tan x - 2 = 0$, so $(2\tan x + 1)(\tan x - 2) = 0$.
   • $\tan x = 2$ or $\tan x = -\dfrac{1}{2}$.
   • List one solution in each relevant quadrant inside the interval.