Reciprocal and inverse trigonometric functionsEdexcel A-Level Maths: Subtopic test

1. 1
   The angle $\theta$ is acute and $\tan\theta=\frac{5}{12}$.

   (a)

   Find the value of $\sec\theta$.

   [1 mark]

   • A$\frac{12}{13}$
   • B$\frac{13}{5}$
   • C$\frac{13}{12}$
   • D$\frac{12}{5}$

   (b)

   Find the value of $\mathrm{cosec}\,\theta$.

   [1 mark]

   • A$\frac{5}{13}$
   • B$\frac{13}{5}$
   • C$\frac{13}{12}$
   • D$\frac{12}{5}$

   (c)

   Use the identity $\mathrm{cosec}^2\theta=1+\cot^2\theta$ to confirm your value of $\mathrm{cosec}\,\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The inverse trigonometric functions $\arcsin$, $\arccos$ and $\arctan$ are defined using their principal values, with angles in radians.

   (a)

   State the domain and range of $y=\arcsin x$.

   [1 mark]

   • ADomain $-\frac{\pi}{2}\le x\le\frac{\pi}{2}$; range $-1\le y\le1$
   • BDomain $-1\le x\le1$; range $0\le y\le\pi$
   • CDomain all real $x$; range $-\frac{\pi}{2}<y<\frac{\pi}{2}$
   • DDomain $-1\le x\le1$; range $-\frac{\pi}{2}\le y\le\frac{\pi}{2}$

   (b)

   Find the exact value of $\arccos\left(-\frac12\right)$.

   [1 mark]

   • A$\frac{2\pi}{3}$
   • B$\frac{\pi}{3}$
   • C$-\frac{\pi}{3}$
   • D$\frac{4\pi}{3}$

   (c)

   Find the exact value of $\arcsin\left(\sin\frac{5\pi}{6}\right)$, explaining why it is not $\frac{5\pi}{6}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In this question $0\le\theta<360^\circ$.

   (a)

   Solve $\mathrm{cosec}\frac{\theta}{2}=2$.

   [3 marks]

   (b)

   Solve $\tan^2\theta+\sec\theta=1$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question you may use the identities $\sec^2\theta\equiv1+\tan^2\theta$ and $\mathrm{cosec}^2\theta\equiv1+\cot^2\theta$.

   (a)

   (i) Prove that $\cot\theta+\tan\theta\equiv\sec\theta\,\mathrm{cosec}\,\theta$.
   (ii) Prove that $\frac{1}{1-\sin\theta}+\frac{1}{1+\sin\theta}\equiv2\sec^2\theta$.

   [6 marks]

   (b)

   Solve $\cot^2x+\mathrm{cosec}\,x=5$ for $0\le x<2\pi$, giving non-exact answers to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions