1.05l Double angle formulae: and compound angle formulae

9.

1. Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\)
2. Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$
   1. find \(\cos x\) in terms of \(k\)
   2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)
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5.

1. Use the substitution \(t = \tan x\) to show that the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$5 t ^ { 4 } - 24 t ^ { 2 } - 5 = 0$$
2. Hence solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation $$12 \tan 2 x + 5 \cot x \sec ^ { 2 } x = 0$$ Show each stage of your working and give your answers to one decimal place.
   

7.

1. Prove that $$\frac { \sin 2 x } { \cos x } + \frac { \cos 2 x } { \sin x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } n \in \mathbb { Z }$$
2. Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$7 + \frac { \sin 4 \theta } { \cos 2 \theta } + \frac { \cos 4 \theta } { \sin 2 \theta } = 3 \cot ^ { 2 } 2 \theta$$ giving your answers in radians to 3 significant figures where appropriate.
   

2.

1. Show that the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ can be written in the form $$\sin 2 \theta = k$$ where \(k\) is a constant to be found.
2. Hence find the smallest positive solution of the equation $$8 \cos \theta = 3 \operatorname { cosec } \theta$$ giving your answer, in degrees, to one decimal place.

9. In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < x \leqslant \pi\), the equation $$2 \sec ^ { 2 } x - 3 \tan x = 2$$ giving the answers, as appropriate, to 3 significant figures.
2. Prove that $$\frac { \sin 3 \theta } { \sin \theta } - \frac { \cos 3 \theta } { \cos \theta } \equiv 2$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Prove that $$\cot ^ { 2 } x - \tan ^ { 2 } x \equiv 4 \cot 2 x \operatorname { cosec } 2 x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }\) $$4 \cot 2 \theta \operatorname { cosec } 2 \theta = 2 \tan ^ { 2 } \theta$$ giving your answers to 2 decimal places.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that the equation $$\frac { 3 \sin \theta \cos \theta } { \cos \theta + \sin \theta } = ( 2 + \sec 2 \theta ) ( \cos \theta - \sin \theta )$$ can be written in the form $$3 \sin 2 \theta - 4 \cos 2 \theta = 2$$
2. Hence solve for \(\pi < x < \frac { 3 \pi } { 2 }\) $$\frac { 3 \sin x \cos x } { \cos x + \sin x } = ( 2 + \sec 2 x ) ( \cos x - \sin x )$$ giving the answer to 3 significant figures.

2.

1. Show that $$\frac { 1 - \cos 2 x } { 2 \sin 2 x } \equiv k \tan x \quad x \neq ( 90 n ) ^ { \circ } \quad n \in \mathbb { Z }$$ where \(k\) is a constant to be found.
2. Hence solve, for \(0 < \theta < 90 ^ { \circ }\) $$\frac { 9 ( 1 - \cos 2 \theta ) } { 2 \sin 2 \theta } = 2 \sec ^ { 2 } \theta$$ giving your answers to one decimal place.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)
   

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 < x < \pi\) $$( x - 2 ) ( \sqrt { 3 } \sec x + 2 ) = 0$$
2. Solve, for \(0 < \theta < 360 ^ { \circ }\) $$10 \sin \theta = 3 \cos 2 \theta$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \equiv \operatorname { cosec } x \quad x \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 < \theta < \frac { \pi } { 2 }\) $$\left( \frac { \cos 2 \theta } { \sin \theta } + \frac { \sin 2 \theta } { \cos \theta } \right) ^ { 2 } = 6 \cot \theta - 4$$ giving your answers to 3 significant figures as appropriate.
3. Using the result from part (a), or otherwise, find the exact value of $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \frac { \cos 2 x } { \sin x } + \frac { \sin 2 x } { \cos x } \right) \cot x d x$$

4. $$f ( x ) = 8 \sin x \cos x + 4 \cos ^ { 2 } x - 3$$

1. Write \(\mathrm { f } ( x )\) in the form $$a \sin 2 x + b \cos 2 x + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
2. Use the answer to part (a) to write \(\mathrm { f } ( x )\) in the form $$R \sin ( 2 x + \alpha ) + c$$ where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 significant figures.
3. Hence, or otherwise,
   1. state the maximum value of \(\mathrm { f } ( x )\)
   2. find the second smallest positive value of \(x\) at which a maximum value of \(\mathrm { f } ( x )\) occurs. Give your answer to 3 significant figures.

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

Solutions relying entirely on calculator technology are not acceptable.

1. Given that $$\sqrt { 2 } \sin \left( x + 45 ^ { \circ } \right) = \cos \left( x - 60 ^ { \circ } \right)$$ show that $$\tan x = - 2 - \sqrt { 3 }$$
2. Hence or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$\sqrt { 2 } \sin ( 2 \theta ) = \cos \left( 2 \theta - 105 ^ { \circ } \right)$$

1. Solve, for \(0 \leqslant x < 360 ^ { \circ }\), the equation

$$2 \cos 2 x = 7 \cos x$$ giving your solutions to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

5.

1. Show that $$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$
2. Hence find, using algebraic integration, $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 3 } x d x$$

4. In this question you should show detailed reasoning. \section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Show that the equation $$2 \sin \left( \theta - 30 ^ { \circ } \right) = 5 \cos \theta$$ can be written in the form $$\tan \theta = 2 \sqrt { 3 }$$
2. Hence, or otherwise, solve for \(0 \leqslant x \leqslant 360 ^ { \circ }\) $$2 \sin \left( x - 10 ^ { \circ } \right) = 5 \cos \left( x + 20 ^ { \circ } \right)$$ giving your answers to one decimal place.

6.

1. The curve \(C _ { 1 }\) has equation $$y = 3 \ln \left( x ^ { 2 } - 5 \right) - 4 x ^ { 2 } + 15 \quad x > \sqrt { 5 }$$ Show that \(C _ { 1 }\) has a stationary point at \(x = \frac { \sqrt { p } } { 2 }\) where \(p\) is a constant to be found.
2. A different curve \(C _ { 2 }\) has equation $$y = 4 x - 12 \sin ^ { 2 } x$$
   1. Show that, for this curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = A + B \sin 2 x$$ where \(A\) and \(B\) are constants to be found.
   2. Hence, state the maximum gradient of this curve.

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = ( 1 + 2 \cos 2 x ) ^ { 2 }$$

1. Show that $$( 1 + 2 \cos 2 x ) ^ { 2 } \equiv p + q \cos 2 x + r \cos 4 x$$ where \(p , q\) and \(r\) are constants to be found. The curve touches the positive \(x\)-axis for the second time when \(x = a\), as shown in Figure 4. The regions bounded by the curve, the \(y\)-axis and the \(x\)-axis up to \(x = a\) are shown shaded in Figure 4.
2. Find, using algebraic integration and making your method clear, the exact total area of the shaded regions. Write your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-32_2255_51_313_1980}

9. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Given that \(\cos 2 \theta - \sin 3 \theta \neq 0\)

1. prove that $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } \equiv \frac { 1 + \sin \theta } { 1 - 2 \sin \theta - 4 \sin ^ { 2 } \theta }$$
2. Hence solve, for \(0 < \theta \leqslant 360 ^ { \circ }\) $$\frac { \cos ^ { 2 } \theta } { \cos 2 \theta - \sin 3 \theta } = 2 \operatorname { cosec } \theta$$ Give your answers to one decimal place. \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-28_2257_52_309_1983}

9.

1. Prove that $$\sec 2 A + \tan 2 A \equiv \frac { \cos A + \sin A } { \cos A - \sin A } \quad A \neq \frac { ( 2 n + 1 ) \pi } { 4 } \quad n \in \mathbb { Z }$$
2. Hence solve, for \(0 \leqslant \theta < 2 \pi\) $$\sec 2 \theta + \tan 2 \theta = \frac { 1 } { 2 }$$ Give your answers to 3 decimal places.

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2. Solve, for \(0 \leqslant x \leqslant 270 ^ { \circ }\), the equation $$\frac { \tan 2 x + \tan 50 ^ { \circ } } { 1 - \tan 2 x \tan 50 ^ { \circ } } = 2$$ Give your answers in degrees to 2 decimal places.
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2. Solve, for \(0 \leqslant \theta < 2 \pi\), $$2 \cos 2 \theta = 5 - 13 \sin \theta$$ Give your answers in radians to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)