1.05l Double angle formulae: and compound angle formulae

2 Solve the equation $$\sin 2 x + 3 \cos 2 x = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1 Solve the equation \(\sin 2 x = 2 \cos 2 x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

3 The reflex angle \(\theta\) is such that \(\cos \theta = k\), where \(0 < k < 1\).

1. Find an expression, in terms of \(k\), for
   1. \(\sin \theta\),
   2. \(\tan \theta\).
   3. Explain why \(\sin 2 \theta\) is negative for \(0 < k < 1\).

11 \includegraphics[max width=\textwidth, alt={}, center]{8da9e73a-3126-471b-b904-25e3c156f6bf-4_519_560_260_797} In the diagram, \(O A B\) is a sector of a circle with centre \(O\) and radius \(r\). The point \(C\) on \(O B\) is such that angle \(A C O\) is a right angle. Angle \(A O B\) is \(\alpha\) radians and is such that \(A C\) divides the sector into two regions of equal area.

1. Show that \(\sin \alpha \cos \alpha = \frac { 1 } { 2 } \alpha\). It is given that the solution of the equation in part (i) is \(\alpha = 0.9477\), correct to 4 decimal places.
2. Find the ratio perimeter of region \(O A C\) : perimeter of region \(A C B\), giving your answer in the form \(k : 1\), where \(k\) is given correct to 1 decimal place.
3. Find angle \(A O B\) in degrees. {www.cie.org.uk} after the live examination series. }

2 By first expanding \(\sin \left( \theta + 30 ^ { \circ } \right)\), solve the equation \(\sin \left( \theta + 30 ^ { \circ } \right) \operatorname { cosec } \theta = 2\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

3

1. Show that \(( \sec x + \cos x ) ^ { 2 }\) can be expressed as \(\sec ^ { 2 } x + a + b \cos 2 x\), where \(a\) and \(b\) are constants to be determined.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \sec x + \cos x ) ^ { 2 } \mathrm {~d} x\).

3 Solve the equation \(\sin \left( 2 \theta + 30 ^ { \circ } \right) = 5 \cos \left( 2 \theta + 60 ^ { \circ } \right)\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

4

1. Find the exact value of \(\int _ { 0 } ^ { 2 } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x\).
2. Find \(\int \left( \tan ^ { 2 } x + 4 \sin ^ { 2 } 2 x \right) \mathrm { d } x\).

3 \includegraphics[max width=\textwidth, alt={}, center]{ed12a4fb-e3bf-4d00-ad09-9ba5be941dd5-04_531_739_258_703} The diagram shows the curve with equation \(y = 3 \sin x - 3 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\). The curve meets the \(x\)-axis at the origin and at the points with \(x\)-coordinates \(a\) and \(\pi\).

1. Find the exact value of \(a\).
2. Find the area of the shaded region.

8

1. Express \(3 \sin 2 \theta \sec \theta + 10 \cos \left( \theta - 30 ^ { \circ } \right)\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(3 \sin 4 \beta \sec 2 \beta + 10 \cos \left( 2 \beta - 30 ^ { \circ } \right) = 2\) for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6

1. Show that \(4 \sin \left( \theta + \frac { 1 } { 3 } \pi \right) \cos \left( \theta - \frac { 1 } { 3 } \pi \right) \equiv \sqrt { 3 } + 2 \sin 2 \theta\).
2. Find the exact value of \(4 \sin \frac { 17 } { 24 } \pi \cos \frac { 1 } { 24 } \pi\).
3. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 8 } \pi } 4 \sin \left( 2 x + \frac { 1 } { 3 } \pi \right) \cos \left( 2 x - \frac { 1 } { 3 } \pi \right) \mathrm { d } x\).

4

1. Show that \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta + 8 \tan \theta + 1 } { 1 - \tan ^ { 2 } \theta }\).
2. Hence solve the equation \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) = 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

1 Solve the equation \(2 \sin \left( \theta + 30 ^ { \circ } \right) + 5 \cos \theta = 2 \sin \theta\) for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{78a9b100-c3bd-4054-b539-ec8304440063-10_551_641_260_751} The diagram shows part of the curve with equation $$y = 4 \sin ^ { 2 } x + 8 \sin x + 3 ,$$ where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(A\) and the shaded region is bounded by the curve and the lines \(x = 0\) and \(y = 0\).

1. Find the exact \(x\)-coordinate of \(A\).
2. Find the exact gradient of the curve at \(A\).
3. Find the exact area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Express \(5 \sqrt { 3 } \cos x + 5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. As \(x\) varies, find the least possible value of $$4 + 5 \sqrt { 3 } \cos x + 5 \sin x$$ and determine the corresponding value of \(x\) where \(- \pi < x < \pi\).
3. Find \(\int \frac { 1 } { ( 5 \sqrt { 3 } \cos 3 \theta + 5 \sin 3 \theta ) ^ { 2 } } d \theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4

1. Show that \(\sin 2 \theta \cot \theta - \cos 2 \theta \equiv 1\).
2. Hence find the exact value of \(\sin \frac { 1 } { 6 } \pi \cot \frac { 1 } { 12 } \pi\).
3. Find the smallest positive value of \(\theta\) (in radians) satisfying the equation $$\sin 2 \theta \cot \theta - 3 \cos 2 \theta = 1 .$$

6 It is given that \(3 \sin 2 \theta = \cos \theta\) where \(\theta\) is an angle such that \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).

1. Find the exact value of \(\sin \theta\).
2. Find the exact value of \(\sec \theta\).
3. Find the exact value of \(\cos 2 \theta\).

7

1. By first expanding \(\cos ( 2 \theta + \theta )\), show that \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\).
2. Find the exact value of \(2 \cos ^ { 3 } \left( \frac { 5 } { 18 } \pi \right) - \frac { 3 } { 2 } \cos \left( \frac { 5 } { 18 } \pi \right)\).
3. Find \(\int \left( 12 \cos ^ { 3 } x - 4 \cos ^ { 3 } 3 x \right) \mathrm { d } x\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7

1. Prove that \(4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \equiv \sqrt { 3 } - \sqrt { 3 } \cos 2 x + \sin 2 x\).
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 5 } { 6 } \pi } 4 \sin x \sin \left( x + \frac { 1 } { 6 } \pi \right) \mathrm { d } x\).
3. Find the smallest positive value of \(y\) satisfying the equation $$4 \sin ( 2 y ) \sin \left( 2 y + \frac { 1 } { 6 } \pi \right) = \sqrt { 3 } .$$ Give your answer in an exact form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 It is given that \(\sec \theta = \sqrt { 17 }\) where \(0 < \theta < \frac { 1 } { 2 } \pi\).
Find the exact value of \(\tan \left( \theta + \frac { 1 } { 4 } \pi \right)\).

8 The expression \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = 12 \sin \theta \cos \theta + 16 \cos ^ { 2 } \theta\).

1. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( 2 \theta - \alpha ) + k\), where \(R > 0,0 < \alpha < \frac { 1 } { 2 } \pi\) and \(k\) is a constant. State the values of \(R\) and \(k\), and give the value of \(\alpha\) correct to 4 significant figures.
2. Find the smallest positive value of \(\theta\) satisfying the equation \(\mathrm { f } ( \theta ) = 17\).
3. Find \(\int f ( \theta ) d \theta\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 It is given that \(\theta\) is an acute angle in degrees such that \(\sin \theta = \frac { 2 } { 3 }\).
Find the exact value of \(\sin \left( \theta + 60 ^ { \circ } \right)\).

6

1. Show that \(\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \equiv 4 + 6 \cos \theta - 4 \cos ^ { 2 } \theta\).
2. Solve the equation $$\operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) + 3 = 0$$ for \(- \pi < \theta < 0\).
3. Find \(\int \operatorname { cosec } \theta \left( 3 \sin 2 \theta + 4 \sin ^ { 3 } \theta \right) \mathrm { d } \theta\).

2 Solve the equation \(\sec \theta \cos \left( \theta - 60 ^ { \circ } \right) = 4\) for \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\).

7

1. Prove that \(\sin 2 x ( \cot x + 3 \tan x ) \equiv 4 - 2 \cos 2 x\).
2. Hence find the exact value of \(\cot \frac { 1 } { 12 } \pi + 3 \tan \frac { 1 } { 12 } \pi\).
3. \includegraphics[max width=\textwidth, alt={}, center]{b104e2a7-06c8-4e2e-a4f9-5095ad56897a-13_796_789_278_708} The diagram shows the curve with equation \(y = 4 - 2 \cos 2 x\), for \(0 < x < 2 \pi\). At the point \(A\), the gradient of the curve is 4 . The point \(B\) is a minimum point. The \(x\)-coordinates of \(A\) and \(B\) are \(a\) and \(b\) respectively. Show that \(\int _ { a } ^ { b } ( 4 - 2 \cos 2 x ) \mathrm { d } x = 3 \pi + 1\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.