1.05l Double angle formulae: and compound angle formulae

7

1. Prove that \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta + 60 ^ { \circ } \right) \equiv \frac { 1 } { 4 } \sqrt { 3 } - \frac { 1 } { 2 } \sin 2 \theta\).
2. Solve the equation \(5 \cos \left( 2 \alpha + 30 ^ { \circ } \right) \cos \left( 2 \alpha + 60 ^ { \circ } \right) = 1\) for \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
3. Show that the exact value of \(\cos 20 ^ { \circ } \cos 50 ^ { \circ } + \cos 40 ^ { \circ } \cos 70 ^ { \circ }\) is \(\frac { 1 } { 2 } \sqrt { 3 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{dcc483e9-630e-4f02-ad8c-4a27c0720fc6-14_2714_38_109_2010}

7

1. Express \(4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right)\) in the form $$a + R \sin ( 2 \theta - \alpha ) ,$$ where \(a\) and \(R\) are positive integers and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-13_2723_33_99_21}
2. Hence find the smallest positive value of \(\theta\) satisfying the equation $$\frac { 1 } { 5 } + 4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right) = 0 .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-14_2714_38_109_2010}

4

1. Show that the equation $$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$ can be written in the form $$3 \tan ^ { 2 } x - 10 \tan x + 3 = 0$$
2. Hence solve the equation $$\tan \left( 45 ^ { \circ } + x \right) = 4 \tan \left( 45 ^ { \circ } - x \right)$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).

7

1. By expanding \(\cos ( 2 x + x )\), show that $$\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x$$
2. Hence, or otherwise, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }$$

7

1. By expanding \(\sin ( 2 x + x )\) and using double-angle formulae, show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 3 } x d x = \frac { 5 } { 24 }$$

2

1. Prove the identity $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos x$$
2. Hence solve the equation $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).

6

1. Express \(\cos ^ { 2 } x\) in terms of \(\cos 2 x\).
2. Hence show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 2 } x \mathrm {~d} x = \frac { 1 } { 6 } \pi + \frac { 1 } { 8 } \sqrt { } 3$$
3. By using an appropriate trigonometrical identity, deduce the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } x \mathrm {~d} x .$$

4 The parametric equations of a curve are $$x = 4 \sin \theta , \quad y = 3 - 2 \cos 2 \theta$$ where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\). Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\), simplifying your answer as far as possible.

3

1. Show that the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\) can be written in the form $$6 \tan ^ { 2 } x - 5 \tan x + 1 = 0$$
2. Hence solve the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

8

1. Prove the identity $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
2. Hence solve the equation $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

8

1. Prove that \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta\).
2. Hence
   1. solve for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\) the equation \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3\),
   2. find the exact value of \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }\).

4

1. Given that \(35 + \sec ^ { 2 } \theta = 12 \tan \theta\), find the value of \(\tan \theta\).
2. Hence, showing the use of an appropriate formula in each case, find the exact value of
   1. \(\tan \left( \theta - 45 ^ { \circ } \right)\),
   2. \(\tan 2 \theta\).

7

1. Show that \(( 2 \sin x + \cos x ) ^ { 2 }\) can be written in the form \(\frac { 5 } { 2 } + 2 \sin 2 x - \frac { 3 } { 2 } \cos 2 x\).
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( 2 \sin x + \cos x ) ^ { 2 } \mathrm {~d} x\).

7

1. Show that \(\tan ^ { 2 } x + \cos ^ { 2 } x \equiv \sec ^ { 2 } x + \frac { 1 } { 2 } \cos 2 x - \frac { 1 } { 2 }\) and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \tan ^ { 2 } x + \cos ^ { 2 } x \right) d x$$
2. \includegraphics[max width=\textwidth, alt={}, center]{48ab71ff-c37b-4e0b-b031-d99b0cf517a8-3_550_785_1573_721} The region enclosed by the curve \(y = \tan x + \cos x\) and the lines \(x = 0 , x = \frac { 1 } { 4 } \pi\) and \(y = 0\) is shown in the diagram. Find the exact volume of the solid produced when this region is rotated completely about the \(x\)-axis.

3

1. Show that \(12 \sin ^ { 2 } x \cos ^ { 2 } x \equiv \frac { 3 } { 2 } ( 1 - \cos 4 x )\).
2. Hence show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 12 \sin ^ { 2 } x \cos ^ { 2 } x d x = \frac { \pi } { 8 } + \frac { 3 \sqrt { } 3 } { 16 }$$

8

1. Prove the identity $$\frac { 1 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } \equiv \operatorname { cosec } x$$
2. Hence solve the equation $$\frac { 2 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } = 3 \cot ^ { 2 } x - 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{3b217eb4-3bd3-4800-a913-749754bf109f-3_401_586_817_778} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\).
2. Show that the equation of the curve can be written $$y = 5 + 8 \sin x - 2 \cos 2 x$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

4

1. Show that \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos \theta\).
2. Hence
   1. find the exact value of \(\sin 105 ^ { \circ } + \sin 165 ^ { \circ }\),
   2. solve the equation \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) = \sec \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

8 \includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve \(( 1,4 )\) and \(( 3,3 )\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).

1 Prove the identity $$\cot \theta - \tan \theta \equiv 2 \cot 2 \theta$$

5 The equation of a curve is \(y = 2 \cos x + \sin 2 x\). Find the \(x\)-coordinates of the stationary points on the curve for which \(0 < x < \pi\), and determine the nature of each of these stationary points.

1

1. Show that the equation $$\sin \left( x - 60 ^ { \circ } \right) - \cos \left( 30 ^ { \circ } - x \right) = 1$$ can be written in the form \(\cos x = k\), where \(k\) is a constant.
2. Hence solve the equation, for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

5

1. Prove the identity $$\sin ^ { 2 } \theta \cos ^ { 2 } \theta \equiv \frac { 1 } { 8 } ( 1 - \cos 4 \theta )$$
2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \sin ^ { 2 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$

6

1. Prove the identity $$\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$$
2. Hence solve the equation $$\cos 4 \theta + 4 \cos 2 \theta = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

4

1. Show that the equation \(\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right)\) can be written in the form $$\tan ^ { 2 } \theta + ( 6 \sqrt { } 3 ) \tan \theta - 5 = 0$$
2. Hence, or otherwise, solve the equation $$\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right) ,$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).