1.05l Double angle formulae: and compound angle formulae

1. (i) Use an appropriate double angle formula to show that

$$\operatorname { cosec } 2 x = \lambda \operatorname { cosec } x \sec x$$ and state the value of the constant \(\lambda\).
(ii) Solve, for \(0 \leqslant \theta < 2 \pi\), the equation $$3 \sec ^ { 2 } \theta + 3 \sec \theta = 2 \tan ^ { 2 } \theta$$ You must show all your working. Give your answers in terms of \(\pi\).

2. Given that \(\tan 40 ^ { \circ } = p\), find in terms of \(p\)

1. \(\cot 40 ^ { \circ }\)
2. \(\sec 40 ^ { \circ }\)
3. \(\tan 85 ^ { \circ }\)

3. Given that $$2 \cos ( x + 50 ) ^ { \circ } = \sin ( x + 40 ) ^ { \circ }$$

1. Show, without using a calculator, that $$\tan x ^ { \circ } = \frac { 1 } { 3 } \tan 40 ^ { \circ }$$
2. Hence solve, for \(0 \leqslant \theta < 360\), $$2 \cos ( 2 \theta + 50 ) ^ { \circ } = \sin ( 2 \theta + 40 ) ^ { \circ }$$ giving your answers to 1 decimal place.

3. (i) (a) Show that \(2 \tan x - \cot x = 5 \operatorname { cosec } x\) may be written in the form $$a \cos ^ { 2 } x + b \cos x + c = 0$$ stating the values of the constants \(a , b\) and \(c\).
(b) Hence solve, for \(0 \leqslant x < 2 \pi\), the equation $$2 \tan x - \cot x = 5 \operatorname { cosec } x$$ giving your answers to 3 significant figures.
(ii) Show that $$\tan \theta + \cot \theta \equiv \lambda \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ stating the value of the constant \(\lambda\).

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\), with equation \(y = 6 \cos x + 2.5 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\)

1. Express \(6 \cos x + 2.5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) to 3 decimal places.
2. Find the coordinates of the points on the graph where the curve \(C\) crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by $$H = 12 + 6 \cos \left( \frac { 2 \pi t } { 52 } \right) + 2.5 \sin \left( \frac { 2 \pi t } { 52 } \right) , \quad 0 \leqslant t \leqslant 52$$ where \(H\) is the number of hours of daylight and \(t\) is the number of weeks since her first recording. Use this function to find
3. the maximum and minimum values of \(H\) predicted by the model,
4. the values for \(t\) when \(H = 16\), giving your answers to the nearest whole number.
   [0pt] [You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.] \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-14_40_58_2460_1893}

1. Given that

$$\tan \theta ^ { \circ } = p , \text { where } p \text { is a constant, } p \neq \pm 1$$ use standard trigonometric identities, to find in terms of \(p\),

1. \(\tan 2 \theta ^ { \circ }\)
2. \(\cos \theta ^ { \circ }\)
3. \(\cot ( \theta - 45 ) ^ { \circ }\) Write each answer in its simplest form.

3. $$g ( \theta ) = 4 \cos 2 \theta + 2 \sin 2 \theta$$ Given that \(\mathrm { g } ( \theta ) = R \cos ( 2 \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),

1. find the exact value of \(R\) and the value of \(\alpha\) to 2 decimal places.
2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$4 \cos 2 \theta + 2 \sin 2 \theta = 1$$ giving your answers to one decimal place. Given that \(k\) is a constant and the equation \(\mathrm { g } ( \theta ) = k\) has no solutions,
3. state the range of possible values of \(k\).

1. (i) Using the identity for \(\tan ( A \pm B )\), solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\),

$$\frac { \tan 2 x + \tan 32 ^ { \circ } } { 1 - \tan 2 x \tan 32 ^ { \circ } } = 5$$ Give your answers, in degrees, to 2 decimal places.
(ii) (a) Using the identity for \(\tan ( A \pm B )\), show that $$\tan \left( 3 \theta - 45 ^ { \circ } \right) \equiv \frac { \tan 3 \theta - 1 } { 1 + \tan 3 \theta } , \quad \theta \neq ( 60 n + 45 ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 < \theta < 180 ^ { \circ }\), $$( 1 + \tan 3 \theta ) \tan \left( \theta + 28 ^ { \circ } \right) = \tan 3 \theta - 1$$

5. (a) Using the formulae $$\begin{gathered} \sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B \\ \cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B \end{gathered}$$ show that

1. \(\sin ( A + B ) - \sin ( A - B ) = 2 \cos A \sin B\),
2. \(\cos ( A - B ) - \cos ( A + B ) = 2 \sin A \sin B\).
   (b) Use the above results to show that $$\frac { \sin ( A + B ) - \sin ( A - B ) } { \cos ( A - B ) - \cos ( A + B ) } = \cot A$$ Using the result of part (b) and the exact values of \(\sin 60 ^ { \circ }\) and \(\cos 60 ^ { \circ }\),
   (c) find an exact value for \(\cot 75 ^ { \circ }\) in its simplest form.
   

   5. continued

   Leave blank

3. (a) Using the identity for \(\cos ( A + B )\), prove that \(\cos \theta \equiv 1 - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).
(b) Prove that \(1 + \sin \theta - \cos \theta \equiv 2 \sin \left( \frac { 1 } { 2 } \theta \right) \left[ \cos \left( \frac { 1 } { 2 } \theta \right) + \sin \left( \frac { 1 } { 2 } \theta \right) \right]\).
(c) Hence, or otherwise, solve the equation $$1 + \sin \theta - \cos \theta = 0 , \quad 0 \leq \theta < 2 \pi$$

6. $$f ( \theta ) = 4 \cos ^ { 2 } \theta - 3 \sin ^ { 2 } \theta$$

1. Show that \(f ( \theta ) = \frac { 1 } { 2 } + \frac { 7 } { 2 } \cos 2 \theta\).
2. Hence, using calculus, find the exact value of \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \theta \mathrm { f } ( \theta ) \mathrm { d } \theta\).

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Use de Moivre's theorem to show that $$\cos 5 x \equiv \cos x \left( a \sin ^ { 4 } x + b \sin ^ { 2 } x + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\cos 5 \theta = \sin 2 \theta \sin \theta - \cos \theta$$ giving your answers to 3 decimal places.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Use De Moivre's theorem to show that $$\cos 6 \theta \equiv 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$
2. Hence determine the smallest positive root of the equation $$48 x ^ { 6 } - 72 x ^ { 4 } + 27 x ^ { 2 } - 1 = 0$$ giving your answer to 3 decimal places.

6. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 16 \sin 2 x } { ( 3 + 4 \sin x ) ^ { 2 } } \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 6 }\) Using the substitution \(u = 3 + 4 \sin x\), show that the area of \(R\) can be written in the form \(a + \ln b\), where \(a\) and \(b\) are rational constants to be found.

4. \includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-3_535_1027_276_577} The diagram above shows a sketch of the curve \(C\) with polar equation $$r = 4 \sin \theta \cos ^ { 2 } \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$ The tangent to \(C\) at the point \(P\) is perpendicular to the initial line.

1. Show that \(P\) has polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right)\). The point \(Q\) on \(C\) has polar coordinates \(\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)\).
   The shaded region \(R\) is bounded by \(O P , O Q\) and \(C\), as shown in the diagram above.
2. Show that the area of \(R\) is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \sin ^ { 2 } 2 \theta \cos 2 \theta + \frac { 1 } { 2 } - \frac { 1 } { 2 } \cos 4 \theta \right) \mathrm { d } \theta$$
3. Hence, or otherwise, find the area of \(R\), giving your answer in the form \(a + b \pi\), where \(a\) and \(b\) are rational numbers.
   (Total 14 marks)

11. (a) Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$ (b) Express \(32 \cos ^ { 6 } \theta\) in the form \(p \cos 6 \theta + q \cos 4 \theta + r \cos 2 \theta + \mathrm { s }\), where \(p , q , r\) and \(s\) are integers.
(c) Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \cos ^ { 6 } \theta \mathrm {~d} \theta$$

3. Find the general solution of the differential equation $$\sin x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y \cos x = \sin 2 x \sin x$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

8. (a) Find the value of \(\lambda\) for which \(y = \lambda x \sin 5 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ (b) Using your answer to part (a), find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 25 y = 3 \cos 5 x$$ Given that at \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5\),
(c) find the particular solution of this differential equation, giving your solution in the form \(y = \mathrm { f } ( x )\).
(d) Sketch the curve with equation \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant \pi\).

3. (a) By writing \(\frac { \pi } { 12 } = \frac { \pi } { 3 } - \frac { \pi } { 4 }\), show that

1. \(\sin \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } - \sqrt { 2 } )\)
2. \(\cos \left( \frac { \pi } { 12 } \right) = \frac { 1 } { 4 } ( \sqrt { 6 } + \sqrt { 2 } )\) (b) Hence find the exact values of \(z\) for which $$z ^ { 4 } = 4 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$$ Give your answers in the form \(z = a + i b\) where \(a , b \in \mathbb { R }\)

6. (a) Use de M oivre's Theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta .$$ (b) Hence or otherwise, prove that the only real solutions of the equation $$\sin 5 \theta = 5 \sin \theta ,$$ are given by \(\theta = n \tau\), where \(n\) is an integer.

$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$ Given that \(\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),

1. find the value of \(R\) and the value of \(\alpha\).
2. Hence solve the equation $$7 \cos 2 x - 24 \sin 2 x = 12.5$$ for \(0 \leqslant x < 180 ^ { \circ }\), giving your answers to 1 decimal place.
3. Express \(14 \cos ^ { 2 } x - 48 \sin x \cos x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\), and \(c\) are constants to be found.
4. Hence, using your answers to parts (a) and (c), deduce the maximum value of $$14 \cos ^ { 2 } x - 48 \sin x \cos x$$

7. Given that $$I _ { n } = \int \frac { \sin n x } { \sin x } \mathrm {~d} x , \quad n \geqslant 1$$

1. prove that, for \(n \geqslant 3\) $$I _ { n } - I _ { n - 2 } = \int 2 \cos ( n - 1 ) x \mathrm {~d} x$$
2. Hence, showing each step of your working, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 6 } } \frac { \sin 5 x } { \sin x } d x$$ giving your answer in the form \(\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )\), where \(a\), \(b\) and \(c\) are integers to be found.

8

1. Use de Moivre's theorem to find an expression for \(\tan 4 \theta\) in terms of \(\tan \theta\).
2. Deduce that \(\cot 4 \theta = \frac { \cot ^ { 4 } \theta - 6 \cot ^ { 2 } \theta + 1 } { 4 \cot ^ { 3 } \theta - 4 \cot \theta }\).
3. Hence show that one of the roots of the equation \(x ^ { 2 } - 6 x + 1 = 0\) is \(\cot ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).
4. Hence find the value of \(\operatorname { cosec } ^ { 2 } \left( \frac { 1 } { 8 } \pi \right) + \operatorname { cosec } ^ { 2 } \left( \frac { 3 } { 8 } \pi \right)\), justifying your answer.

5

1. Use de Moivre's theorem to prove that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1 .$$
2. Hence find the largest positive root of the equation $$64 x ^ { 6 } - 96 x ^ { 4 } + 36 x ^ { 2 } - 3 = 0 ,$$ giving your answer in trigonometrical form.

1. (i) Find the exact value of \(x\) such that

$$3 \tan ^ { - 1 } ( x - 2 ) + \pi = 0$$ (ii) Solve, for \(- \pi < \theta < \pi\), the equation $$\cos 2 \theta - \sin \theta - 1 = 0$$ giving your answers in terms of \(\pi\).