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

1. 1
   The angle $\theta$ is acute and $\sin\theta=\frac35$.

   (a)

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

   [1 mark]

   • A$\frac35$
   • B$\frac54$
   • C$\frac53$
   • D$\frac43$

   (b)

   Find the value of $\cot\theta$.

   [1 mark]

   • A$\frac43$
   • B$\frac34$
   • C$\frac54$
   • D$\frac53$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The inverse functions arcsin, arccos and arctan are defined by restricting sine, cosine and tangent to intervals on which each is one-to-one, so each takes its principal value.

   (a)

   State the range of $\arcsin x$.

   [1 mark]

   • A$0\le y\le\pi$
   • B$-\frac{\pi}{2}<y<\frac{\pi}{2}$
   • C$-1\le y\le1$
   • D$-\frac{\pi}{2}\le y\le\frac{\pi}{2}$

   (b)

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

   [1 mark]

   • A$\frac{\pi}{3}$
   • B$\frac{2\pi}{3}$
   • C$-\frac{\pi}{6}$
   • D$\frac{4\pi}{3}$

   (c)

   Find the exact value of $\arctan\left(-\sqrt3\right)+\arccos\left(\frac{\sqrt3}{2}\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $2\sec^2\theta=5-\tan\theta$ for $0\le\theta<2\pi$.

   (a)

   Show that the equation can be written as $2\tan^2\theta+\tan\theta-3=0$.

   [3 marks]

   (b)

   Hence solve the equation, giving non-exact answers in radians to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The reciprocal functions are $\sec\theta=\frac{1}{\cos\theta}$, $\operatorname{cosec}\theta=\frac{1}{\sin\theta}$ and $\cot\theta=\frac{1}{\tan\theta}$. Give angles in radians for $0<\theta<2\pi$.

   (a)

   (i) Prove that $\tan\theta+\cot\theta\equiv\sec\theta\operatorname{cosec}\theta$.
   (ii) Hence find the exact value of $\sec\theta\operatorname{cosec}\theta$ when $\tan\theta=3$.

   [6 marks]

   (b)

   Solve $\operatorname{cosec}^2\theta-3\cot\theta-1=0$, giving non-exact answers to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions