Inverse trigonometric functionsEdexcel International A Level Maths: Subtopic test

1. 1
   The inverse trigonometric functions $\arcsin$, $\arccos$ and $\arctan$ are defined by their principal values, with angles in radians.

   (a)

   What is the range of $\arcsin x$?

   [1 mark]

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

   (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 $\arctan\left(-\sqrt3\right)+\arctan(1)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\alpha=\arcsin\left(\frac{4}{5}\right)$, where $\alpha$ is measured in degrees.

   (a)

   Find the exact value of $\cos\alpha$.

   [1 mark]

   • A$\frac45$
   • B$\frac53$
   • C$-\frac35$
   • D$\frac35$

   (b)

   Find the exact value of $\tan\alpha$.

   [1 mark]

   • A$\frac34$
   • B$\frac43$
   • C$\frac45$
   • D$\frac54$

   (c)

   Find $\arcsin(\sin150^{\circ})$ and explain why it is not $150^{\circ}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Angles are measured in radians and a calculator may be used.

   (a)

   Solve $\tan x=-3$ for $-\pi<x<\pi$, giving your answers to 3 significant figures.

   [3 marks]

   (b)

   The function $g$ is defined by $g(x)=\pi-2\arccos x$ for $-1\le x\le1$.
   (i) Find the range of $g$.
   (ii) Solve $g(x)=\frac{\pi}{2}$, giving an exact answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A vertical mast of height $12$ m stands on level ground. An observer stands $d$ m from the foot of the mast, with eye level at ground level, and $\theta$ is the angle of elevation of the top of the mast. A calculator may be used.

   (a)

   (i) Find $\theta$, in degrees to 1 decimal place, when $d=5$.
   (ii) The observer walks directly towards the mast from $d=16$ to $d=5$. Find the increase in $\theta$, in degrees to 1 decimal place.
   (iii) Find the value of $d$, to 3 significant figures, for which $\theta=30^{\circ}$.

   [6 marks]

   (b)

   A flagpole of height $3$ m is fixed to the top of the mast. The angle that the flagpole subtends at the observer is $\varphi$.
   (i) Show that $\varphi=\arctan\frac{15}{d}-\arctan\frac{12}{d}$.
   (ii) Find $\varphi$ in degrees to 1 decimal place when $d=10$.
   (iii) Explain why $\varphi$ approaches $0^{\circ}$ both as $d\to\infty$ and as $d\to0$.

   [6 marks]

   Total for question 4: 12 marks

End of questions