1.05l Double angle formulae: and compound angle formulae

3.(i)Determine the value of \(k\) such that $$\arctan \frac { 1 } { 2 } - \arctan \frac { 1 } { 3 } = \arctan k$$ (ii)(a)Prove that $$\cos 3 A \equiv 4 \cos ^ { 3 } A - 3 \cos A$$ Given that \(a = \cos 20 ^ { \circ }\) (b)write down,in terms of \(a\) ,an expression for \(\cos 40 ^ { \circ }\) (c)determine,in terms of \(a\) ,a simplified expression for \(\cos 80 ^ { \circ }\) (d)Use part(a)to show that $$4 a ^ { 3 } - 3 a = \frac { 1 } { 2 }$$ (e)Hence,or otherwise,show that $$\cos 20 ^ { \circ } \cos 40 ^ { \circ } \cos 80 ^ { \circ } = \frac { 1 } { 8 }$$

2.Solve,for \(0 \leqslant x \leqslant 360 ^ { \circ }\) $$\sin 47 ^ { \circ } \cos ^ { 3 } x + \cos 47 ^ { \circ } \sin x \cos ^ { 2 } x = \frac { 1 } { 2 } \cos ^ { 2 } x$$

8

1. Show that \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = \sin 2 \theta\).
2. In this question you must show detailed reasoning. Solve \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta } = 3 \cos 2 \theta\) for \(0 \leq \theta \leq \pi\).

3 In this question you must show detailed reasoning. Given that \(5 \sin 2 x = 3 \cos x\), where \(0 ^ { \circ } < x < 90 ^ { \circ }\), find the exact value of \(\sin x\).

8 In this question you must show detailed reasoning. The diagram shows triangle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-06_737_1383_456_342} The angles \(C A B\) and \(A B C\) are each \(45 ^ { \circ }\), and angle \(A C B = 90 ^ { \circ }\).
The points \(D\) and \(E\) lie on \(A C\) and \(A B\) respectively. \(A E = D E = 1 , D B = 2\). Angle \(B E D = 90 ^ { \circ }\), angle \(E B D = 30 ^ { \circ }\) and angle \(D B C = 15 ^ { \circ }\).

1. Show that \(B C = \frac { \sqrt { 2 } + \sqrt { 6 } } { 2 }\).
2. By considering triangle \(B C D\), show that \(\sin 15 ^ { \circ } = \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }\).

14 Some students are trying to prove an identity for \(\sin ( A + B )\). They start by drawing two right-angled triangles \(O D E\) and \(O E F\), as shown. \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-22_695_662_477_689} The students' incomplete proof continues,
Let angle \(D O E = A\) and angle \(E O F = B\).
In triangle OFR,
Line \(1 \quad \sin ( A + B ) = \frac { R F } { O F }\) Line 2 $$= \frac { R P + P F } { O F }$$ Line 3 $$= \frac { D E } { O F } + \frac { P F } { O F } \text { since } D E = R P$$ Line 4 $$= \frac { D E } { \cdots \cdots } \times \frac { \cdots \cdots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 \(=\) \(\_\_\_\_\) \(+ \cos A \sin B\) 14

1. Explain why \(\frac { P F } { E F } \times \frac { E F } { O F }\) in Line 4 leads to \(\cos A \sin B\) in Line 5
   14
2. Complete Line 4 and Line 5 to prove the identity Line 4 $$= \frac { D E } { \ldots \ldots } \times \frac { \cdots \ldots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 = \(+ \cos A \sin B\) 14
3. Explain why the argument used in part (a) only proves the identity when \(A\) and \(B\) are acute angles. 14
4. Another student claims that by replacing \(B\) with \(- B\) in the identity for \(\sin ( A + B )\) it is possible to find an identity for \(\sin ( A - B )\). Assuming the identity for \(\sin ( A + B )\) is correct for all values of \(A\) and \(B\), prove a similar result for \(\sin ( A - B )\).

8

1. Given that $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ show that $$8 \cot ^ { 2 } \theta - 2 \cot \theta - 1 = 0$$ 8
2. Hence, solve $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ in the interval \(0 < \theta < 2 \pi\) Give your answers to two decimal places.
   8
3. Solve $$9 \sin ^ { 2 } \left( 2 x - \frac { \pi } { 4 } \right) + \sin \left( 4 x - \frac { \pi } { 2 } \right) = 8$$ in the interval \(0 < x < \frac { \pi } { 2 }\) Give your answers to one decimal place.

15

1. Show that $$\sin x - \sin x \cos 2 x \approx 2 x ^ { 3 }$$ for small values of \(x\).
   15
2. Hence, show that the area between the graph with equation $$y = \sqrt { 8 ( \sin x - \sin x \cos 2 x ) }$$ the positive \(x\)-axis and the line \(x = 0.25\) can be approximated by $$\text { Area } \approx 2 ^ { m } \times 5 ^ { n }$$ where \(m\) and \(n\) are integers to be found.
   15
3. (i) Explain why $$\int _ { 6.3 } ^ { 6.4 } 2 x ^ { 3 } \mathrm {~d} x$$ is not a suitable approximation for $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ Question 15 continues on the next page 15 (c) (ii) Explain how $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ may be approximated by $$\int _ { a } ^ { b } 2 x ^ { 3 } \mathrm {~d} x$$ for suitable values of \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-31_2492_1721_217_150}
   \includegraphics[max width=\textwidth, alt={}]{042e248a-9efa-4844-957d-f05715900ffc-36_2486_1719_221_150}

4 Given that \(\theta\) is a small angle, find an approximation for \(\cos 2 \theta\) Circle your answer. \(1 - \frac { \theta ^ { 2 } } { 2 }\) \(2 - 2 \theta ^ { 2 }\) \(1 - 2 \theta ^ { 2 }\) \(1 - \theta ^ { 2 }\)

9 A robotic arm which is attached to a flat surface at the origin \(O\), is used to draw a graphic design. The arm is made from two rods \(O P\) and \(P Q\), each of length \(d\), which are joined at \(P\).
A pen is attached to the arm at \(Q\).
The coordinates of the pen are controlled by adjusting the angle \(O P Q\) and the angle \(\theta\) between \(O P\) and the \(x\)-axis. For this particular design the pen is made to move so that the two angles are always equal to each other with \(0 \leq \theta \leq \frac { \pi } { 2 }\) as shown in Figure 2. \begin{figure}[h] \end{figure} 9

1. Show that the \(x\)-coordinate of the pen can be modelled by the equation $$x = d \left( \cos \theta + \sin \left( 2 \theta - \frac { \pi } { 2 } \right) \right)$$ 9
2. Hence, show that $$x = d \left( 1 + \cos \theta - 2 \cos ^ { 2 } \theta \right)$$ 9
3. It can be shown that $$x = \frac { 9 d } { 8 } - d \left( \cos \theta - \frac { 1 } { 4 } \right) ^ { 2 }$$ State the greatest possible value of \(x\) and the corresponding value of \(\cos \theta\) 9
4. Figure 3 below shows the arm when the \(x\)-coordinate is at its greatest possible value. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-14_570_773_456_630}
   \end{figure} Find, in terms of \(d\), the exact distance \(O Q\). \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-15_2488_1716_219_153}

18 Two particles, \(P\) and \(Q\), are projected at the same time from a fixed point \(X\), on the ground, so that they travel in the same vertical plane. \(P\) is projected at an acute angle \(\theta ^ { \circ }\) to the horizontal, with speed \(u \mathrm {~ms} ^ { - 1 }\) \(Q\) is projected at an acute angle \(2 \theta ^ { \circ }\) to the horizontal, with speed \(2 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Both particles land back on the ground at exactly the same point, \(Y\).
Resistance forces to motion may be ignored.
18

1. Show that $$\cos 2 \theta = \frac { 1 } { 8 }$$ 18
2. \(\quad P\) takes a total of 0.4 seconds to travel from \(X\) to \(Y\).
   Find the time taken by \(Q\) to travel from \(X\) to \(Y\).
   18
3. State one modelling assumption you have chosen to make in this question.
   [0pt] [1 mark]
   

   19

   Two skaters, Jo and Amba, are separately skating across a smooth, horizontal surface of ice. Both are moving in the same direction, so that their paths are straight and are parallel to each other. Jo is moving with constant velocity \(( 2.8 \mathbf { i } + 9.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At time \(t = 0\) seconds Amba is at position ( \(2 \mathbf { i } - 7 \mathbf { j }\) ) metres and is moving with a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Explain why Amba's velocity must be in the form \(k ( 2.8 \mathbf { i } + 9.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

2 In this question you must show detailed reasoning.
Solve the equation \(2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0\) giving each answer in the form \(\ln ( p + q \sqrt { r } )\) where \(p\) and \(q\) are rational numbers, and \(r\) is an integer, whose values are to be determined. You are given that the matrix \(\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)\), where \(a\) is a positive constant, represents the transformation R which is a reflection in 3-D.

1. State the plane of reflection of \(R\).
2. Determine the value of \(a\).
3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.
   1. By using Euler's formula show that \(\cosh ( \mathrm { iz } ) = \cos z\).
   2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.] A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255} In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$
   3. 1. Write down the general solution to (\textit{).
      2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic. In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$ where \(\lambda\) is a positive constant.
   4. In the case where \(\lambda = 16\) the door is held open at an angle of 0.9 radians and then released from rest at time \(t = 0\).
      1. Find, in a real form, the general solution of ( \(\dagger\) ).
      2. Find the particular solution of ( \(\dagger\) ).
      3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( \(\dagger\) ) improves the model.
      4. Find the value of \(\lambda\) for which the door is critically damped.

4 In this question you must show detailed reasoning.

1. By writing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta ) .$$
2. Hence show that \(\sin \frac { 1 } { 8 } \pi = \frac { 1 } { 2 } \sqrt [ 6 ] { 20 - 14 \sqrt { 2 } }\).
   1. Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
   2. By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).

10

1. Prove that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$ and hence show that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
2. Hence solve the equation \(\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

9

1. Prove that \(\operatorname { cosec } 2 x - \cot 2 x \equiv \tan x\) and hence find an exact value for \(\tan \left( \frac { 3 } { 8 } \pi \right)\).
2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 3 } { 8 } \pi } ( \operatorname { cosec } 2 x - \cot 2 x ) ^ { 2 } \mathrm {~d} x\).

12

1. Use the identity \(\tan 2 x \equiv \frac { 2 \tan x } { 1 - \tan ^ { 2 } x }\) to show that \(\tan 4 x \equiv \frac { 4 \left( 1 - \tan ^ { 2 } x \right) \tan x } { 1 - 6 \tan ^ { 2 } x + \tan ^ { 4 } x }\).
2. Hence, given that \(x = \frac { 1 } { 16 } \pi\) is a root of the equation \(\tan ^ { 4 } x + p \tan ^ { 3 } x - 6 \tan ^ { 2 } x - p \tan x + 1 = 0\) where \(p\) is a positive constant, find the value of \(p\).

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 \(\cot \theta\).

10

1. Use de Moivre's theorem to show that \(2 \cos 6 \theta \equiv 64 \cos ^ { 6 } \theta - 96 \cos ^ { 4 } \theta + 36 \cos ^ { 2 } \theta - 2\).
2. Hence find, in exact trigonometric form, the six roots of the equation $$x ^ { 6 } - 6 x ^ { 4 } + 9 x ^ { 2 } - 3 = 0$$
3. By considering the product of these six roots, determine the exact value of $$\cos \left( \frac { 1 } { 18 } \pi \right) \cos \left( \frac { 5 } { 18 } \pi \right) \cos \left( \frac { 7 } { 18 } \pi \right) .$$

10

1. Show that \(\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta\).
2. Hence solve the equation \(\sin 3 \theta = \cos 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
3. Show that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\). Hence, by writing \(\cos 2 \theta - \sin 3 \theta\) in terms of \(\sin \theta\), use your answer to part (ii) to determine the solutions of \(4 x ^ { 3 } - 2 x ^ { 2 } - 3 x + 1 = 0\).

11

1. Prove that $$\sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \frac { 1 } { 2 } \sin ^ { 2 } \theta - \frac { 3 } { 4 } = \frac { 1 } { 4 } \sqrt { 3 } \sin 2 \theta .$$
2. Hence solve the equation $$2 \sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \sin ^ { 2 } \theta = 1 \text { for } - \pi \leqslant \theta \leqslant \pi .$$

12 For each positive integer \(n\), the function \(\mathrm { F } _ { n }\) is defined for all real angles \(\theta\) by $$\mathrm { F } _ { n } ( \theta ) = c ^ { 2 n } + s ^ { 2 n }$$ where \(c = \cos \theta\) and \(s = \sin \theta\).

1. Prove the identity $$\mathrm { F } _ { n + 2 } ( \theta ) - \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta \times \mathrm { F } _ { n + 1 } ( \theta ) \equiv \mathrm { F } _ { n + 3 } ( \theta )$$ Let \(z\) denote the complex number \(c + \mathrm { i } s\).
2. Using de Moivre's theorem,
   1. express \(z + z ^ { - 1 }\) and \(z - z ^ { - 1 }\) in terms of \(c\) and \(s\) respectively,
   2. prove the identity \(8 \left( c ^ { 6 } + s ^ { 6 } \right) \equiv 3 \cos 4 \theta + 5\) and deduce that $$c ^ { 6 } + s ^ { 6 } \equiv \cos ^ { 2 } 2 \theta + \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta$$
   3. Prove by induction that, for all positive integers \(n\), $$c ^ { 2 n + 4 } + s ^ { 2 n + 4 } \leqslant \cos ^ { 2 } 2 \theta + \frac { 1 } { 2 ^ { n + 1 } } \sin ^ { 2 } 2 \theta$$ [You are given that the range of the function \(\mathrm { F } _ { n }\) is \(\frac { 1 } { 2 ^ { n - 1 } } \leqslant \mathrm {~F} _ { n } ( \theta ) \leqslant 1\).] {www.cie.org.uk} after the live examination series. }

4 Solve the equation \(\sin 2 x = \sqrt { 3 } \cos x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

9 In this question, \(x\) denotes an angle measured in degrees.

1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).

a) Given that \(\alpha\) and \(\beta\) are two angles such that \(\tan \alpha = 2 \cot \beta\), show that $$\tan ( \alpha + \beta ) = - ( \tan \alpha + \tan \beta )$$ b) Find all values of \(\theta\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\) satisfying the equation $$4 \tan \theta = 3 \sec ^ { 2 } \theta - 7$$