1.05a Sine, cosine, tangent: definitions for all arguments

8. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the function \(\mathrm { h } ( x )\) with equation $$h ( x ) = 45 + 15 \sin x + 21 \sin \left( \frac { x } { 2 } \right) + 25 \cos \left( \frac { x } { 2 } \right) \quad x \in [ 0,40 ]$$

1. Show that $$\frac { \mathrm { d } h } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 6 t - 17 \right) \left( 9 t ^ { 2 } + 4 t - 3 \right) } { 2 \left( 1 + t ^ { 2 } \right) ^ { 2 } }$$ where \(t = \tan \left( \frac { x } { 4 } \right)\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a52911da-4b69-4d86-975e-d10e3a481e1d-16_581_1403_1263_331} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Source: \({ } ^ { 1 }\) Data taken on 29th December 2016 from \href{http://www.ukho.gov.uk/easytide/EasyTide}{http://www.ukho.gov.uk/easytide/EasyTide} Figure 2 shows a graph of predicted tide heights, in metres, for Portland harbour from 08:00 on the 3rd January 2017 to the end of the 4th January \(2017 { } ^ { 1 }\). The graph of \(k \mathrm {~h} ( x )\), where \(k\) is a constant and \(x\) is the number of hours after 08:00 on 3rd of January, can be used to model the predicted tide heights, in metres, for this period of time.
2. 1. Suggest a value of \(k\) that could be used for the graph of \(k \mathrm {~h} ( x )\) to form a suitable model.
   2. Why may such a model be suitable to predict the times when the tide heights are at their peaks, but not to predict the heights of these peaks?
3. Use Figure 2 and the result of part (a) to estimate, to the nearest minute, the time of the highest tide height on the 4th January 2017.

3 In this question you must show detailed reasoning. You are given that \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\).
Explain why \(\tan 690 ^ { \circ } = - \frac { 1 } { \sqrt { 3 } }\).

4 The diagram shows part of the graph of \(y = \cos x\), where \(x\) is measured in radians. \includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-05_609_846_294_607}

1. Use the copy of this diagram in the Printed Answer Booklet to find an approximate solution to the equation \(x = \cos x\).
2. Use an iterative method to find the solution to the equation \(x = \cos x\) correct to 3 significant figures. You should show your first, second and last two iterations, writing down all the figures on your calculator.

12 One of the rides at a theme park is a room where the floor and ceiling both move up and down for \(10 \pi\) seconds. At time \(t\) seconds after the ride begins, the distance \(f\) metres of the floor above the ground is $$f = 1 - \cos t$$ At time \(t\) seconds after the ride begins, the distance \(c\) metres of the ceiling above the ground is $$c = 8 - 4 \sin t$$ The ride is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-16_448_766_932_635} 12

1. Show that the initial distance between the floor and ceiling is 8 metres.
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}]{6a03a035-ff32-4734-864b-a076aa9cbec0-17_2500_1721_214_148}

7. \begin{figure}[h] \end{figure} Figure 3 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = - 4 \cos x$$ where \(x\) is measured in radians.
Points \(P\) and \(Q\) lie on the curve and are shown in Figure 3.

1. State
   1. the coordinates of \(P\)
   2. the coordinates of \(Q\) The curve \(C _ { 2 }\) has equation \(y = - 4 \cos x + k\) where \(x\) is measured in radians and \(k\) is a constant. Given that \(C _ { 2 }\) has a maximum \(y\) value of 11
2. 1. state the value of \(k\)
   2. state the coordinates of the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate. On the opposite page there is a copy of Figure 3 labelled Diagram 1.
3. Using Diagram 1, state the number of solutions of the equation $$- 4 \cos x = 5 - \frac { 10 } { \pi } x$$ giving a reason for your answer. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-23_860_1016_1676_529} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

2 Sketch the curve with equation \(y = \tan x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
On the same diagram, sketch the curve with equation \(y = \tan ^ { - 1 } x\) for all \(x\).
State the geometrical relationship between the curves.

6 Given that the angle \(\theta\) is acute and \(\cos \theta = \frac { 3 } { 4 }\) find, without using a calculator, the exact value of \(\sin 2 \theta\) and of \(\cot \theta\).

12 Points \(A\) and \(B\) lie on a line of greatest slope of a plane inclined at an angle \(\alpha\) to the horizontal, with \(B\) above \(A\). A particle is projected from \(A\) with speed \(u\) at an angle \(\theta\) to the plane and subsequently strikes the plane at right angles at \(B\).

1. Show that \(2 \tan \alpha \tan \theta = 1\).
2. In either order, show that
   1. the vertical height of \(B\) above \(A\) is \(\frac { 2 u ^ { 2 } \tan ^ { 2 } \alpha } { g \left( 1 + 4 \tan ^ { 2 } \alpha \right) }\),
   2. the time of flight from \(A\) to \(B\) is \(\frac { 2 u \sec \alpha } { g \sqrt { 1 + 4 \tan ^ { 2 } \alpha } }\).

\includegraphics{figure_1} The diagram shows the curve with equation \(y = a\sin(bx) + c\) for \(0 \leqslant x \leqslant 2\pi\), where \(a\), \(b\) and \(c\) are positive constants.

1. State the values of \(a\), \(b\) and \(c\). [3]
2. For these values of \(a\), \(b\) and \(c\), determine the number of solutions in the interval \(0 \leqslant x \leqslant 2\pi\) for each of the following equations:
   1. \(a\sin(bx) + c = 7 - x\) [1]
   2. \(a\sin(bx) + c = 2\pi(x - 1)\). [1]

Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for

1. \(\cos \theta\), [1]
2. \(\tan \theta\), [2]
3. \(\sin(\theta + \pi)\). [1]

The function f is such that \(\mathrm{f}(x) = a + b \cos x\) for \(0 \leqslant x \leqslant 2\pi\). It is given that \(\mathrm{f}\left(\frac{1}{3}\pi\right) = 5\) and \(\mathrm{f}(\pi) = 11\).

1. Find the values of the constants \(a\) and \(b\). [3]
2. Find the set of values of \(k\) for which the equation \(\mathrm{f}(x) = k\) has no solution. [3]

A particle \(P\) is projected with speed \(u \text{ m s}^{-1}\) at an angle \(\tan^{-1} 2\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. When \(P\) has travelled a distance \(56 \text{ m}\) horizontally from \(O\), it is at a vertical height \(H \text{ m}\) above the plane. When \(P\) has travelled a distance \(84 \text{ m}\) horizontally from \(O\), it is at a vertical height \(\frac{1}{2}H \text{ m}\) above the plane. Find, in either order, the value of \(u\) and the value of \(H\). [5]

A particle \(P\) is projected with speed \(u\) at an angle \(\alpha\) above the horizontal from a point \(O\) on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements of \(P\) from \(O\) at a time \(t\) are denoted by \(x\) and \(y\) respectively.

1. Derive the equation of the trajectory of \(P\) in the form $$y = x \tan \alpha - \frac{gx^2}{2u^2} \sec^2 \alpha.$$ [3]
2. The greatest height of \(P\) above the plane is denoted by \(H\). When \(P\) is at a height of \(\frac{3}{4}H\), it is travelling at a horizontal distance \(d\). Given that \(\tan \alpha = 3\) and in terms of \(H\), the two possible values of \(d\). [6]

\includegraphics{figure_1} Two smooth uniform spheres \(A\) and \(B\) of equal radius have masses 2 kg and 1 kg respectively. They are moving on a smooth horizontal plane when they collide. Immediately before the collision the speed of \(A\) is 2.5 m s\(^{-1}\) and the speed of \(B\) is 1.3 m s\(^{-1}\). When they collide the line joining their centres makes an angle \(\alpha\) with the direction of motion of \(A\) and an angle \(\beta\) with the direction of motion of \(B\), where \(\tan \alpha = \frac{4}{3}\) and \(\tan \beta = \frac{12}{5}\) as shown in Fig. 1.

1. Find the components of the velocities of \(A\) and \(B\) perpendicular and parallel to the line of centres immediately before the collision. [4]

The coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{2}\).

1. Find, to one decimal place, the speed of each sphere after the collision. [9]

\includegraphics{figure_4} Mary swims in still water at 0.85 m s\(^{-1}\). She swims across a straight river which is 60 m wide and flowing at 0.4 m s\(^{-1}\). She sets off from a point \(A\) on the near bank and lands at a point \(B\), which is directly opposite \(A\) on the far bank, as shown in Fig. 4. Find

1. the angle between the near bank and the direction in which Mary swims, [3]
2. the time she takes to cross the river. [3]

\includegraphics{figure_5} A little further downstream a large tree has fallen from the far bank into the river. The river is modelled as flowing at 0.5 m s\(^{-1}\) for a width of 40 m from the near bank, and 0.2 m s\(^{-1}\) for the 20 m beyond this. Nassim swims at 0.85 m s\(^{-1}\) in still water. He swims across the river from a point \(C\) on the near bank. The point \(D\) on the far bank is directly opposite \(C\), as shown in Fig. 5. Nassim swims at the same angle to the near bank as Mary.

1. Find the maximum distance, downstream from \(CD\), of Nassim during the crossing. [5]
2. Show that he will land at the point \(D\). [4]

Two smooth uniform spheres \(A\) and \(B\) have equal radii. Sphere \(A\) has mass \(m\) and sphere \(B\) has mass \(km\). The spheres are at rest on a smooth horizontal table. Sphere \(A\) is then projected along the table with speed \(u\) and collides with \(B\). Immediately before the collision, the direction of motion of \(A\) makes an angle of \(60°\) with the line joining the centres of the two spheres. The coefficient of restitution between the spheres is \(\frac{1}{2}\).

1. Show that the speed of \(B\) immediately after the collision is \(\frac{3u}{4(k + 1)}\). [6] Immediately after the collision the direction of motion of \(A\) makes an angle arctan \((2\sqrt{3})\) with the direction of motion of \(B\).
2. Show that \(k = \frac{1}{2}\). [6]
3. Find the loss of kinetic energy due to the collision. [4]

A small smooth ball of mass \(\frac{1}{2}\) kg is falling vertically. The ball strikes a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{1}{3}\). Immediately before striking the plane the ball has speed 10 m s\(^{-1}\). The coefficient of restitution between ball and plane is \(\frac{1}{2}\). Find

1. the speed, to 3 significant figures, of the ball immediately after the impact, [5]
2. the magnitude of the impulse received by the ball as it strikes the plane. [2]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation $$y = \sin (\cos x).$$ The curve cuts the \(x\)-axis at the points \(A\) and \(C\) and the \(y\)-axis at the point \(B\).

1. Find the coordinates of the points \(A\), \(B\) and \(C\). [3]
2. Prove that \(B\) is a stationary point. [2]

Given that the region \(OCB\) is convex,

1. show that, for \(0 \leq x \leq \frac{\pi}{2}\), $$\sin (\cos x) \leq \cos x$$ and $$(1 - \frac{2}{\pi} x) \sin 1 \leq \sin (\cos x)$$ and state in each case the value or values of \(x\) for which equality is achieved. [6]
2. Hence show that $$\frac{\pi}{4} \sin 1 < \int_0^{\frac{\pi}{2}} \sin(\cos x) \, dx < 1.$$ [4]

State the interval for which \(\sin x\) is a decreasing function for \(0° \leq x \leq 360°\) [2 marks]

It is given that \(\tan \theta^\circ = k\), where \(k\) is a constant. Find \(\tan (\theta + 180)^\circ\) Circle your answer. [1 mark] \(-k\) \qquad \(-\frac{1}{k}\) \qquad \(\frac{1}{k}\) \qquad \(k\)

The monthly mean temperature of a city, \(T\) degrees Celsius, may be modelled by the equation $$T = 15 + 8 \sin (30m - 120)^\circ$$ where \(m\) is the month number, counting January = 1, February = 2, through to December = 12

1. Using this model, calculate the monthly mean temperature of the city for May, the fifth month. [2 marks]
2. Using this model, find the month with the highest mean temperature. [2 marks]
3. Climate change may affect the parameters, 8, 30, 120 and 15, used in this model.
   1. State, with a reason, which parameter would be increased because of an overall rise in temperatures. [1 mark]
   2. State, with a reason, which parameter would be increased because of the occurrence of more extreme temperatures. [1 mark]