Trigonometry (A2)AQA A-Level Maths: Topic test

1. 1
   A sector of a circle has radius $9$ cm and angle $140^\circ$.

   (a)

   What is the angle of the sector in radians?

   [1 mark]

   • A$\frac{7\pi}{9}$
   • B$\frac{7\pi}{18}$
   • C$\frac{14\pi}{9}$
   • D$\frac{9\pi}{7}$

   (b)

   What is the length of the arc of the sector?

   [1 mark]

   • A$\frac{63\pi}{2}$ cm
   • B$\frac{7\pi}{2}$ cm
   • C$7\pi$ cm
   • D$1260$ cm

   (c)

   Find the perimeter of the sector, giving your answer in exact form.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In this question $x$ is small and measured in radians, so the standard small angle approximations may be used.

   (a)

   Which expression is approximately equal to $\tan2x+\sin x$?

   [1 mark]

   • A$2x$
   • B$3x$
   • C$x$
   • D$2x^2$

   (b)

   Which expression is approximately equal to $\cos3x$?

   [1 mark]

   • A$1-\frac{3x^2}{2}$
   • B$1-9x^2$
   • C$1-\frac{x^2}{2}$
   • D$1-\frac{9x^2}{2}$

   (c)

   Find the value approached by $\dfrac{x\sin3x}{1-\cos x}$ as $x\to0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The function $f$ is defined by $f(x)=\arctan(2x)+\frac{\pi}{4}$ for $x\in\mathbb{R}$.

   (a)

   State the range of $f$ and find the exact value of $f\left(\frac{\sqrt3}{2}\right)$.

   [3 marks]

   (b)

   Find $f^{-1}(x)$ and state its domain.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=8\sin x\cos x+3\cos2x$, where $x$ is in radians.

   (a)

   Express $f(x)$ in the form $R\sin(2x+\alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$. Hence find the maximum value of $f(x)$ and the smallest positive value of $x$ at which it occurs.

   [6 marks]

   (b)

   Hence solve $f(x)=2$ for $0\le x\le\pi$, giving your answers to 2 decimal places.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   In this question give exact values, using the fact that $105^\circ=60^\circ+45^\circ$.

   (a)

   What is the exact value of $\sin105^\circ$?

   [1 mark]

   • A$\frac{\sqrt6-\sqrt2}{4}$
   • B$\frac{\sqrt2-\sqrt6}{4}$
   • C$\frac{\sqrt3+1}{2}$
   • D$\frac{\sqrt6+\sqrt2}{4}$

   (b)

   What is the exact value of $\cos105^\circ$?

   [1 mark]

   • A$\frac{\sqrt2-\sqrt6}{4}$
   • B$\frac{\sqrt6-\sqrt2}{4}$
   • C$\frac{\sqrt6+\sqrt2}{4}$
   • D$\frac{\sqrt3-1}{2}$

   (c)

   Show that $\tan105^\circ=-2-\sqrt3$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $g$ is defined by $g(x)=2\sec x$ for $0\le x\le2\pi$, where $x\ne\frac{\pi}{2}$ and $x\ne\frac{3\pi}{2}$.

   (a)

   Which statement gives the range of $g$?

   [1 mark]

   • A$-2\le g(x)\le2$
   • B$g(x)\le-1$ or $g(x)\ge1$
   • C$g(x)\le-2$ or $g(x)\ge2$
   • D$g(x)\in\mathbb{R}$

   (b)

   How many solutions does the equation $g(x)=3$ have in the domain?

   [1 mark]

   • A$1$
   • B$2$
   • C$3$
   • D$4$

   (c)

   Solve $g(x)=-4$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A pendulum bob hangs on a light string of length $0.8$ m. The bob is pulled aside until the string makes a small angle $\theta$ radians with the vertical, and is released from rest. Take $g=9.8$ m s$^{-2}$ and ignore air resistance.

   (a)

   Show that the height $h$ metres through which the bob is raised, relative to its lowest point, is approximately $0.4\theta^2$.

   [3 marks]

   (b)

   Use conservation of energy to show that the speed of the bob at its lowest point is approximately $2.8\theta$ m s$^{-1}$. Hence find the value of $\theta$ for which this speed is $0.35$ m s$^{-1}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The expression $\cos3\theta$ can be written in terms of $\cos\theta$, where $\theta$ is in radians.

   (a)

   By writing $3\theta=2\theta+\theta$, prove that $\cos3\theta\equiv4\cos^3\theta-3\cos\theta$.

   [6 marks]

   (b)

   Hence solve $8\cos^3\theta-6\cos\theta=1$ for $0\le\theta\le\pi$, giving your answers in terms of $\pi$.

   [6 marks]

   Total for question 8: 12 marks

End of questions