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

1. 1
   No calculator may be used in this question. Let $\theta=\dfrac{5\pi}{6}$, where $\theta$ is in radians.

   (a)

   What is the exact value of $\sin\theta$?

   [1 mark]

   • A$-\dfrac{1}{2}$
   • B$\dfrac{1}{2}$
   • C$\dfrac{\sqrt3}{2}$
   • D$-\dfrac{\sqrt3}{2}$

   (b)

   What is the exact value of $\tan\theta$?

   [1 mark]

   • A$\sqrt3$
   • B$-\sqrt3$
   • C$\dfrac{1}{\sqrt3}$
   • D$-\dfrac{1}{\sqrt3}$

   (c)

   Find the exact value of $\sin\theta+\cos\theta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In triangle $XYZ$, $XY=7$ cm, $XZ=10$ cm and angle $YXZ=60^\circ$.

   (a)

   What is the area of triangle $XYZ$?

   [1 mark]

   • A$30.3\text{ cm}^2$
   • B$35.0\text{ cm}^2$
   • C$17.5\text{ cm}^2$
   • D$60.6\text{ cm}^2$

   (b)

   What is the length of $YZ$?

   [1 mark]

   • A$14.8\text{ cm}$
   • B$12.2\text{ cm}$
   • C$8.89\text{ cm}$
   • D$3.00\text{ cm}$

   (c)

   Find the size of angle $XYZ$, giving your answer to the nearest $0.1^\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The angle $\theta$ is acute and $\cos\theta=\dfrac{4}{5}$.

   (a)

   Find the exact values of $\sin\theta$ and $\tan\theta$.

   [3 marks]

   (b)

   Show that $\dfrac{\sin\theta}{1+\cos\theta}+\dfrac{1+\cos\theta}{\sin\theta}=\dfrac{2}{\sin\theta}$, and hence find its exact value for this $\theta$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In the quadrilateral $WXYZ$, $WX=8$ cm, $XY=11$ cm, $YZ=6$ cm, $WZ=9$ cm and angle $WXY=75^\circ$.

   (a)

   Find the length of $WY$, the size of angle $WZY$ and the area of triangle $WYZ$. Give lengths and areas to 3 significant figures and angles to the nearest $0.1^\circ$.

   [6 marks]

   (b)

   Find the area of the quadrilateral $WXYZ$, and the perpendicular distance from $X$ to the line $WY$. Use your answer to part (a) where appropriate.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the equation $2\cos x=\sqrt3$, where $x$ is in radians and $0\le x\le2\pi$.

   (a)

   How many solutions does the equation have?

   [1 mark]

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

   (b)

   What is the larger solution?

   [1 mark]

   • A$\dfrac{11\pi}{6}$
   • B$\dfrac{5\pi}{6}$
   • C$\dfrac{7\pi}{6}$
   • D$\dfrac{5\pi}{3}$

   (c)

   Solve $2\cos2x=\sqrt3$ for $0\le x\le2\pi$, giving your answers as exact multiples of $\pi$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A student claims that $\sin^2\theta-\cos^2\theta=1-2\cos^2\theta$ for all values of $\theta$.

   (a)

   Which identity is needed to justify the claim?

   [1 mark]

   • A$\tan\theta=\dfrac{\sin\theta}{\cos\theta}$
   • B$\sin(-\theta)=-\sin\theta$
   • C$\cos(\theta+360^\circ)=\cos\theta$
   • D$\sin^2\theta+\cos^2\theta=1$

   (b)

   Which expression equals $\sin^2\theta-\cos^2\theta$ when written in terms of $\sin\theta$ only?

   [1 mark]

   • A$1-2\sin^2\theta$
   • B$2\sin^2\theta-1$
   • C$1$
   • D$\sin^2\theta-1$

   (c)

   Hence solve $\sin^2\theta-\cos^2\theta=-\dfrac{1}{2}$ for $0^\circ\le\theta\le180^\circ$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The equation $3\sin^2x+5\cos x-1=0$ is to be solved, where $x$ is in degrees and $0^\circ\le x<360^\circ$.

   (a)

   Show that the equation can be written as $3\cos^2x-5\cos x-2=0$.

   [3 marks]

   (b)

   Hence solve the equation, giving your answers to the nearest $0.1^\circ$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The depth of water, $h$ metres, in a tidal harbour is modelled by $h=5+3\cos(30t)^\circ$, where $t$ is the time in hours after midnight and $0\le t\le24$. At a second harbour, the depth is modelled over the same period by $h=5+3\sin(30t)^\circ$.

   (a)

   For the first harbour, find the greatest depth and all the times at which it occurs. Find all the times when the depth is 6.5 m.

   [6 marks]

   (b)

   Find all the times when the two harbours have the same depth.

   [6 marks]

   Total for question 8: 12 marks

End of questions