Trigonometry in contextEdexcel A-Level Maths: Subtopic test

1. 1
   The height $h$ metres above the ground of a point $P$ on the rim of a vertical wheel, $t$ seconds after the wheel starts to turn, is modelled by $h=6-5\cos(30t)^\circ$.

   (a)

   Find the height of $P$ above the ground when $t=0$.

   [1 mark]

   • A$6$ m
   • B$1$ m
   • C$11$ m
   • D$5$ m

   (b)

   How long does the wheel take to make one complete turn?

   [1 mark]

   • A$12$ s
   • B$30$ s
   • C$6$ s
   • D$360$ s

   (c)

   Find the first time, to 3 significant figures, at which $P$ is $9$ m above the ground.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of hours of daylight, $D$, in a town on day $n$ of the year ($n=1$ is 1 January) is modelled by $D=12+4\sin\left(\dfrac{2\pi(n-80)}{365}\right)$, where the angle is in radians.

   (a)

   What is the greatest number of hours of daylight predicted by the model?

   [1 mark]

   • A$12$
   • B$4$
   • C$16$
   • D$8$

   (b)

   On which day of the year, to the nearest whole day, does the model predict the longest daylight?

   [1 mark]

   • A$80$
   • B$263$
   • C$354$
   • D$171$

   (c)

   Use the model to find, to the nearest day, the number of days in the year on which there are more than $14$ hours of daylight.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball is kicked from level ground with speed $20\ \text{m s}^{-1}$ at an angle $\alpha$ above the horizontal. Modelling the ball as a particle moving freely under gravity ($g=9.8\ \text{m s}^{-2}$), the horizontal distance it travels before landing is $R=\dfrac{400\sin2\alpha}{g}$ metres.

   (a)

   The ball lands $30$ m away. Find the two possible values of $\alpha$, to 1 decimal place, with $0<\alpha<90^\circ$.

   [3 marks]

   (b)

   The ball is kicked at an angle $\alpha$ where $\tan\alpha=\frac34$. Find the exact value of $\sin2\alpha$ and hence the distance the ball travels, to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The depth of water, $d$ metres, at the end of a pier $t$ hours after midnight is modelled by $d=7+3\sin\left(\dfrac{\pi t}{6}\right)+4\cos\left(\dfrac{\pi t}{6}\right)$, for $0\le t\le12$, where angles are in radians.

   (a)

   (i) Write $3\sin\left(\frac{\pi t}{6}\right)+4\cos\left(\frac{\pi t}{6}\right)$ in the form $R\sin\left(\frac{\pi t}{6}+\alpha\right)$, with $R>0$ and $0<\alpha<\frac\pi2$, giving $\alpha$ to 3 significant figures.
   (ii) Hence find the greatest depth of water and the first time at which it occurs, in hours to 2 decimal places.

   [6 marks]

   (b)

   A boat can enter the harbour only when $d\ge9$.
   (i) Solve $d=9$ for $0\le t\le12$, giving $t$ to 1 decimal place.
   (ii) Find the total time in the interval $0\le t\le12$ for which the boat can enter, to 2 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions