1.05l Double angle formulae: and compound angle formulae

4 The curve with equation \(y = \mathrm { e } ^ { - x } \sin x\) has one stationary point for which \(0 \leqslant x \leqslant \pi\).

1. Find the \(x\)-coordinate of this point.
2. Determine whether this point is a maximum or a minimum point.

5

1. Prove the identity \(\cos 4 \theta - 4 \cos 2 \theta + 3 \equiv 8 \sin ^ { 4 } \theta\).
2. Using this result find, in simplified form, the exact value of $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta$$

4 The angles \(\alpha\) and \(\beta\) lie in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\), and are such that $$\tan \alpha = 2 \tan \beta \quad \text { and } \quad \tan ( \alpha + \beta ) = 3 .$$ Find the possible values of \(\alpha\) and \(\beta\).

3 Solve the equation $$\cos \left( \theta + 60 ^ { \circ } \right) = 2 \sin \theta$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4 It is given that \(\mathrm { f } ( x ) = 4 \cos ^ { 2 } 3 x\).

1. Find the exact value of \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 9 } \pi \right)\).
2. Find \(\int \mathrm { f } ( x ) \mathrm { d } x\).

3 Solve the equation $$\sin \left( \theta + 45 ^ { \circ } \right) = 2 \cos \left( \theta - 30 ^ { \circ } \right)$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

8

1. By first expanding \(\sin ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
2. Show that, after making the substitution \(x = \frac { 2 \sin \theta } { \sqrt { 3 } }\), the equation \(x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0\) can be written in the form \(\sin 3 \theta = \frac { 3 } { 4 }\).
3. Hence solve the equation $$x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0$$ giving your answers correct to 3 significant figures.

4

1. Show that \(\cos \left( \theta - 60 ^ { \circ } \right) + \cos \left( \theta + 60 ^ { \circ } \right) \equiv \cos \theta\).
2. Given that \(\frac { \cos \left( 2 x - 60 ^ { \circ } \right) + \cos \left( 2 x + 60 ^ { \circ } \right) } { \cos \left( x - 60 ^ { \circ } \right) + \cos \left( x + 60 ^ { \circ } \right) } = 3\), find the exact value of \(\cos x\).

3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).

6 The angles \(A\) and \(B\) are such that $$\sin \left( A + 45 ^ { \circ } \right) = ( 2 \sqrt { } 2 ) \cos A \quad \text { and } \quad 4 \sec ^ { 2 } B + 5 = 12 \tan B$$ Without using a calculator, find the exact value of \(\tan ( A - B )\).

5

1. Prove the identity \(\tan 2 \theta - \tan \theta \equiv \tan \theta \sec 2 \theta\).
2. Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \tan \theta \sec 2 \theta \mathrm {~d} \theta = \frac { 1 } { 2 } \ln \frac { 3 } { 2 }\).

3 Express the equation \(\cot 2 \theta = 1 + \tan \theta\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

3 By expressing the equation \(\tan \left( \theta + 60 ^ { \circ } \right) + \tan \left( \theta - 60 ^ { \circ } \right) = \cot \theta\) in terms of \(\tan \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

9

1. By first expanding \(\cos ( 2 x + x )\), show that \(\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x\).
2. Hence solve the equation \(\cos 3 x + 3 \cos x + 1 = 0\), for \(0 \leqslant x \leqslant \pi\).
3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 3 } x \mathrm {~d} x\).

4

1. By first expanding \(\tan ( 2 x + x )\), show that the equation \(\tan 3 x = 3 \cot x\) can be written in the form \(\tan ^ { 4 } x - 12 \tan ^ { 2 } x + 3 = 0\).
2. Hence solve the equation \(\tan 3 x = 3 \cot x\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\).

3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).

1

1. Show that $$\tan ( r + 1 ) - \tan r = \frac { \sin 1 } { \cos ( r + 1 ) \cos r }$$ Let \(\mathrm { u } _ { \mathrm { r } } = \frac { 1 } { \cos ( \mathrm { r } + 1 ) \cos \mathrm { r } }\).
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } u _ { r }\).
3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.

7

1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 5 } { 2 } } = 4 x y \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r = \sin 4 \theta\).
   The curve \(C\) has polar equation \(r = \sin 4 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
2. Sketch \(C\) and state the equation of the line of symmetry.
3. Find the exact value of the area of the region enclosed by \(C\).
4. Using the identity \(\sin 4 \theta \equiv 4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta\), find the maximum distance of \(C\) from the line \(\theta = \frac { 1 } { 2 } \pi\). Give your answer correct to 2 decimal places.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8

1. Show that \(\sin 2 x \cot x \equiv 2 \cos ^ { 2 } x\).
2. Using the identity in part (i),
   1. find the least possible value of $$3 \sin 2 x \cot x + 5 \cos 2 x + 8$$ as \(x\) varies,
   2. find the exact value of \(\int _ { \frac { 1 } { 8 } \pi } ^ { \frac { 1 } { 6 } \pi } \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).

2

1. Given that \(\tan 2 \theta \cot \theta = 8\), show that \(\tan ^ { 2 } \theta = \frac { 3 } { 4 }\).
2. Hence solve the equation \(\tan 2 \theta \cot \theta = 8\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5 The angle \(x\), measured in degrees, satisfies the equation $$\cos \left( x - 30 ^ { \circ } \right) = 3 \sin \left( x - 60 ^ { \circ } \right)$$

1. By expanding each side, show that the equation may be simplified to $$( 2 \sqrt { } 3 ) \cos x = \sin x$$
2. Find the two possible values of \(x\) lying between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
3. Find the exact value of \(\cos 2 x\), giving your answer as a fraction.

3 Find the values of \(x\) satisfying the equation $$3 \sin 2 x = \cos x$$ for \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{d527d21f-0ab5-40fa-8cfd-ebfb4aba0a87-3_493_863_264_641} The diagram shows the part of the curve \(y = \sin ^ { 2 } x\) for \(0 \leqslant x \leqslant \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).
2. Hence find the \(x\)-coordinates of the points on the curve at which the gradient of the curve is 0.5 . [3]
3. By expressing \(\sin ^ { 2 } x\) in terms of \(\cos 2 x\), find the area of the region bounded by the curve and the \(x\)-axis between 0 and \(\pi\).

4

1. Prove the identity $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \tan 2 x .$$
2. Hence solve the equation $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) = 2 ,$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

6

1. Express \(8 \sin \theta - 15 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$8 \sin \theta - 15 \cos \theta = 14$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).