Further trigonometry – Year 13 A level maths worksheet with answers to download as PDF

In this question \(\theta\) is small and measured in radians. Use the approximations \(\sin\theta\approx\theta\), \(\tan\theta\approx\theta\) and \(\cos\theta\approx1-\dfrac{\theta^2}2\).

1. Estimate \(\cos0.06\) and compare with your calculator. [2 marks]
2. Find an approximation for \(\dfrac{1-\cos4\theta}{\theta\sin2\theta}\). [3 marks]
3. Find an approximation for \(\dfrac{\sin3\theta\tan2\theta}{\theta^2}\). [3 marks]

Write down the exact value of each expression. Use a right-angled triangle or the unit circle where necessary.

1. \(\sec60^\circ\) and \(\operatorname{cosec}30^\circ\) [2 marks]
2. \(\cot45^\circ\) and \(\cot150^\circ\) [3 marks]
3. \(\sec120^\circ\) and \(\operatorname{cosec}240^\circ\) [3 marks]

Solve each equation for \(0^\circ\le x<360^\circ\), giving answers to 1 decimal place where necessary.

1. \(\sec x=-2\) [2 marks]
2. \(\tan^2x+\sec x=1\) [5 marks]
3. \(\operatorname{cosec}^2x-\cot x=3\) [5 marks]

Use the compound angle formulae to find the exact value of each expression.

1. \(\sin15^\circ\), using \(15^\circ=45^\circ-30^\circ\) [3 marks]
2. \(\cos75^\circ\) [3 marks]
3. \(\tan105^\circ\), using \(105^\circ=60^\circ+45^\circ\) [4 marks]

Solve each equation for \(0^\circ\le x<360^\circ\).

1. \(\sin2x=\sin x\) [4 marks]
2. \(\cos2x+3\cos x+2=0\) [5 marks]

\(f(x)=5\sin x+12\cos x\).

1. Express \(f(x)\) in the form \(R\sin(x+\alpha)\) where \(R>0\) and \(0^\circ\lt \alpha<90^\circ\). Give \(\alpha\) to 2 decimal places. [3 marks]
2. State the maximum and minimum values of \(f(x)\) and the smallest positive value of \(x\) where the maximum occurs. [3 marks]
3. Solve \(f(x)=6.5\) for \(0^\circ\le x<360^\circ\). [4 marks]

A surveyor stands 400 m from the base of a wind turbine tower on flat ground. She measures the angle of elevation of the top as 0.08 radians.

1. Use a small-angle approximation to estimate the height of the tower. [2 marks]
2. Find the exact height using \(h=400\tan0.08\), to 3 significant figures. [2 marks]
3. Find the percentage error in your estimate. [2 marks]
4. Explain why the approximation would be unsuitable for an angle of 1 radian. [2 marks]

A ramp must lift a wheelchair user a vertical height of \(h=0.45\) m up to a library entrance. The ramp makes an angle \(\theta\) with the horizontal. The ramp length is \(L=h\operatorname{cosec}\theta\) and the horizontal run is \(d=h\cot\theta\).

1. Find \(L\) and \(d\) when \(\theta=5^\circ\). [3 marks]
2. The building regulations in the UK allow a gradient of at most 1 in 12, so \(\tan\theta\le\tfrac1{12}\). Find the largest allowed angle. [2 marks]
3. Hence find the minimum length of the ramp and the horizontal run needed. [3 marks]

The depth of water in a harbour in Cornwall is modelled by \(h=6+3\sin(30t)^\circ+4\cos(30t)^\circ\) metres, where \(t\) is the time in hours after midnight, for \(0\le t\le12\).

1. Express \(3\sin(30t)^\circ+4\cos(30t)^\circ\) in the form \(R\sin(30t+\alpha)^\circ\). [3 marks]
2. State the greatest and least depths. [2 marks]
3. Boats can only enter when the depth is at least 9 m. Find the times when this is possible. [5 marks]

A goalkeeper kicks a football at speed \(u=20\) m s\(^{-1}\) at an angle \(\theta\) above the horizontal. Ignoring air resistance, the horizontal range is \(R=\dfrac{u^2\sin2\theta}g\), taking \(g=9.8\) m s\(^{-2}\).

1. Find the maximum range and the angle at which it occurs. [3 marks]
2. The ball must land 30 m away on a flat pitch. Show that \(\sin2\theta=0.735\). [2 marks]
3. Find the two possible angles of projection. [3 marks]

Two skateboard ramps are joined end to end. The first has a gradient of \(\dfrac13\) (so \(\tan A=\dfrac13\)) and the second a gradient of \(\dfrac12\) (so \(\tan B=\dfrac12\)), where \(A\) and \(B\) are acute angles.

1. Show that \(\tan(A+B)=1\). [3 marks]
2. Hence find the total angle \(A+B\) through which the path has turned upwards. [1 mark]
3. Find \(\sin(A+B)\) exactly. [3 marks]
4. Find \(\tan(B-A)\). [2 marks]

A floodlight is mounted 12 m above a flat training pitch. The beam makes an angle \(\theta\) with the horizontal. The beam length is \(12\operatorname{cosec}\theta\) metres and the distance from the base of the pole to the lit spot is \(12\cot\theta\) metres.

1. Find the beam length and the lit distance when \(\theta=30^\circ\). [3 marks]
2. The groundskeeper wants to light a spot exactly 20 m from the pole. Find \(\theta\). [2 marks]
3. Show that \(\operatorname{cosec}^2\theta-\cot^2\theta=1\) and use it to find the beam length in part (b) without finding \(\theta\). [4 marks]

A swing has chains of length \(L=2.5\) m. When the chains make a small angle \(\theta\) with the vertical, the seat is displaced horizontally by \(L\sin\theta\) and raised by \(L(1-\cos\theta)\).

1. Use small-angle approximations to show that the horizontal displacement is approximately \(L\theta\) and the rise is approximately \(\dfrac{L\theta^2}2\). [2 marks]
2. For \(\theta=0.2\) radians find the approximate values of both quantities. [2 marks]
3. Find the exact values and the percentage errors, to 2 significant figures. [4 marks]

In a physics experiment light passes through a polarising filter. The intensity transmitted is \(I=I_0\cos^2\theta\), where \(\theta\) is the angle between the filter and the polarisation of the light, \(0^\circ\le\theta\le90^\circ\).

1. Show that \(I=\dfrac{I_0}2(1+\cos2\theta)\). [2 marks]
2. Find \(\theta\) when the intensity is \(0.75I_0\). [2 marks]
3. Find \(\theta\) when the intensity is \(0.2I_0\), to 1 decimal place. [2 marks]
4. Explain why \(I\) never exceeds \(I_0\). [1 mark]

Triple-angle formulae are used in engineering to analyse waves. You may use \(\sin2x=2\sin x\cos x\) and \(\cos2x=1-2\sin^2x\).

1. Prove that \(\operatorname{cosec}2x+\cot2x\equiv\cot x\). [4 marks]
2. Prove that \(\sin3x\equiv3\sin x-4\sin^3x\). [4 marks]
3. Hence solve \(\sin3x=\sin x\) for \(0^\circ\le x\le180^\circ\). [4 marks]
4. Use part (a) with \(x=15^\circ\) to show that \(\cot15^\circ=2+\sqrt3\). [3 marks]

A weather station records the temperature \(T\) (°C) \(t\) hours after 11:00. A scientist models it as \(T=14+3\sin(15t)^\circ+4\cos(15t)^\circ\), for \(0\le t\le24\).

1. Verify two of the table values. [2 marks]
2. Write \(3\sin(15t)^\circ+4\cos(15t)^\circ\) in the form \(R\sin(15t+\alpha)^\circ\) and state the maximum and minimum temperature. [4 marks]
3. At what time of day is the temperature greatest? [3 marks]
4. For how long in the 24 hours is the temperature at least 17 °C? [4 marks]

\(\(2\sec^2x+3\tan x-5=0,\qquad 0^\circ\le x<360^\circ.\)\)

1. Show that the equation can be written as \(2\tan^2x+3\tan x-3=0\). [2 marks]
2. Solve the equation, giving the solutions to 1 decimal place. [5 marks]
3. Explain why \(\sec x\) never takes values between \(-1\) and \(1\), using the table and the definition. [2 marks]
4. State how many solutions you expect for \(\tan x=k\) in the given range, and why. [2 marks]

A roof truss has two rafters making acute angles \(A\) and \(B\) with the horizontal, where \(\sin A=\dfrac8{17}\) and \(\cos B=\dfrac35\).

1. Find \(\cos A\) and \(\sin B\) exactly. [2 marks]
2. Show that \(\sin(A+B)=\dfrac{84}{85}\). [3 marks]
3. Find \(\cos(A+B)\) and verify that \(\sin^2(A+B)+\cos^2(A+B)=1\). [3 marks]
4. Find \(\tan(A-B)\) exactly. [4 marks]
5. Find \(A+B\) in degrees to 1 decimal place and say whether the combined angle is acute. [2 marks]

The brightness reading of a monitor beside a solar farm is modelled by \(B(x)=\dfrac1{7+3\sin x+4\cos x}\) where \(x\) is the angle in degrees of the sun through the day, \(0^\circ\le x<360^\circ\).

1. Write \(3\sin x+4\cos x\) in the form \(R\sin(x+\alpha)\). [3 marks]
2. Find the maximum and minimum values of \(B\). [3 marks]
3. Find the value of \(x\) at which \(B\) is greatest. [2 marks]
4. Solve \(B(x)=\dfrac14\). [4 marks]

Let \(\theta=36^\circ\) and \(c=\cos\theta\).

1. Explain why \(3\theta+2\theta=180^\circ\) and deduce that \(\cos3\theta=-\cos2\theta\). [2 marks]
2. Using \(\cos3\theta=4c^3-3c\) and \(\cos2\theta=2c^2-1\), show that \(4c^3+2c^2-3c-1=0\). [2 marks]
3. Show that \((c+1)\) is a factor and hence find the exact value of \(\cos36^\circ\). [4 marks]
4. Find the exact value of \(\cos72^\circ\) and show that \(\cos36^\circ-\cos72^\circ=\dfrac12\). [3 marks]