1.05l Double angle formulae: and compound angle formulae

14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h] \end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.

6 In this question you must show detailed reasoning.

1. Use the formula for \(\tan ( A - B )\) to show that \(\tan \frac { \pi } { 12 } = 2 - \sqrt { 3 }\).
2. Solve the equation \(2 \sqrt { 3 } \sin 3 A - 2 \cos 3 A = 1\) for \(0 ^ { \circ } \leqslant A < 180 ^ { \circ }\).

12

1. By first writing \(\tan 3 \theta\) as \(\tan ( 2 \theta + \theta )\), show that \(\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }\).
2. Hence show that there are always exactly two different values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) which satisfy the equation \(3 \tan 3 \theta = \tan \theta + k\),
   where \(k\) is a non-zero constant. \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

5. (a) Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$ (b) Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$ (c) Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(d) Hence, for \(0 \leq \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
Hence, for \(0 \leq \theta < \pi\), solve \includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.

3 It is given that \(3 \cos \theta - 2 \sin \theta = R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).

1. Find the value of \(R\).
2. Show that \(\alpha \approx 33.7 ^ { \circ }\).
3. Hence write down the maximum value of \(3 \cos \theta - 2 \sin \theta\) and find a positive value of \(\theta\) at which this maximum value occurs.

6

1. Express \(\cos 2 x\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
2. Hence show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } x \mathrm {~d} x = \frac { \pi } { a }\), where \(a\) is an integer.

3

1. Express \(\cos 2 x\) in terms of \(\sin x\).
2. 1. Hence show that \(3 \sin x - \cos 2 x = 2 \sin ^ { 2 } x + 3 \sin x - 1\) for all values of \(x\).
   2. Solve the equation \(3 \sin x - \cos 2 x = 1\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Use your answer from part (a) to find \(\int \sin ^ { 2 } x \mathrm {~d} x\).

7

1. Use the identity $$\tan ( A + B ) = \frac { \tan A + \tan B } { 1 - \tan A \tan B }$$ to express \(\tan 2 x\) in terms of \(\tan x\).
2. Show that $$2 - 2 \tan x - \frac { 2 \tan x } { \tan 2 x } = ( 1 - \tan x ) ^ { 2 }$$ for all values of \(x , \tan 2 x \neq 0\).

7

1. 1. Express \(6 \sin \theta + 8 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(6 \sin 2 x + 8 \cos 2 x = 7\), giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
2. 1. Prove the identity \(\frac { \sin 2 x } { 1 - \cos 2 x } = \frac { 1 } { \tan x }\).
   2. Hence solve the equation $$\frac { \sin 2 x } { 1 - \cos 2 x } = \tan x$$ giving all solutions in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Solve the equation $$5 \sin 2 x + 3 \cos x = 0$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) to the nearest \(0.1 ^ { \circ }\), where appropriate.
3. Given that \(\sin 2 x + \cos 2 x = 1 + \sin x\) and \(\sin x \neq 0\), show that \(2 ( \cos x - \sin x ) = 1\).

6

1. 1. Express \(\sin 2 \theta\) and \(\cos 2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
   2. Given that \(0 < \theta < \frac { \pi } { 2 }\) and \(\cos \theta = \frac { 3 } { 5 }\), show that \(\sin 2 \theta = \frac { 24 } { 25 }\) and find the value of \(\cos 2 \theta\).
2. A curve has parametric equations $$x = 3 \sin 2 \theta , \quad y = 4 \cos 2 \theta$$
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\).
   2. At the point \(P\) on the curve, \(\cos \theta = \frac { 3 } { 5 }\) and \(0 < \theta < \frac { \pi } { 2 }\). Find an equation of the tangent to the curve at the point \(P\).

6

1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
2. Using the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) :
   1. express \(\cos 2 x\) in terms of \(\sin x\) and \(\cos x\);
   2. show, by writing \(3 x\) as \(( 2 x + x )\), that $$\cos 3 x = 4 \cos ^ { 3 } x - 3 \cos x$$
3. Show that \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 3 } x \mathrm {~d} x = \frac { 2 } { 3 }\).

4

1. 1. Express \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
   2. Express \(\cos 2 x\) in terms of \(\cos x\).
2. Show that $$\sin 2 x - \tan x = \tan x \cos 2 x$$ for all values of \(x\).
3. Solve the equation \(\sin 2 x - \tan x = 0\), giving all solutions in degrees in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).

3

1. By writing \(\sin 3 x\) as \(\sin ( x + 2 x )\), show that \(\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x\) for all values of \(x\).
2. Hence, or otherwise, find \(\int \sin ^ { 3 } x \mathrm {~d} x\).

5

1. The angle \(\alpha\) is acute and \(\sin \alpha = \frac { 4 } { 5 }\).
   1. Find the value of \(\cos \alpha\).
   2. Express \(\cos ( \alpha - \beta )\) in terms of \(\sin \beta\) and \(\cos \beta\).
   3. Given also that the angle \(\beta\) is acute and \(\cos \beta = \frac { 5 } { 13 }\), find the exact value of \(\cos ( \alpha - \beta )\).
2. 1. Given that \(\tan 2 x = 1\), show that \(\tan ^ { 2 } x + 2 \tan x - 1 = 0\).
   2. Hence, given that \(\tan 45 ^ { \circ } = 1\), show that \(\tan 22 \frac { 1 } { 2 } ^ { \circ } = \sqrt { 2 } - 1\).

6

1. 1. Show that the equation \(3 \cos 2 x + 7 \cos x + 5 = 0\) can be written in the form \(a \cos ^ { 2 } x + b \cos x + c = 0\), where \(a , b\) and \(c\) are integers.
   2. Hence find the possible values of \(\cos x\).
2. 1. Express \(7 \sin \theta + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(7 \sin \theta + 3 \cos \theta = 4\) for all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving \(\theta\) to the nearest \(0.1 ^ { \circ }\).
3. 1. Given that \(\beta\) is an acute angle and that \(\tan \beta = 2 \sqrt { 2 }\), show that \(\cos \beta = \frac { 1 } { 3 }\).
   2. Hence show that \(\sin 2 \beta = p \sqrt { 2 }\), where \(p\) is a rational number.

4 Solve the equation \(2 \sin 2 \theta + \cos 2 \theta = 1\), for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

3 Given that \(\sin ( \theta + \alpha ) = 2 \sin \theta\), show that \(\tan \theta = \frac { \sin \alpha } { 2 - \cos \alpha }\). Hence solve the equation \(\sin \left( \theta + 40 ^ { \circ } \right) = 2 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

3 Solve the equation \(\cos 2 \theta = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving your answers in terms of \(\pi\).

6 The angle \(x\) radians satisfies the equation $$\cos \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { \sqrt { 2 } }$$

1. Find the general solution of this equation, giving the roots as exact values in terms of \(\pi\).
2. Find the number of roots of the equation which lie between 0 and \(2 \pi\).

7

1. Use the identity \(\tan ( A - B ) = \frac { \tan A - \tan B } { 1 + \tan A \tan B }\) with \(A = ( r + 1 ) x\) and \(B = r x\) to show that $$\tan r x \tan ( r + 1 ) x = \frac { \tan ( r + 1 ) x } { \tan x } - \frac { \tan r x } { \tan x } - 1$$ (4 marks)
2. Use the method of differences to show that $$\tan \frac { \pi } { 50 } \tan \frac { 2 \pi } { 50 } + \tan \frac { 2 \pi } { 50 } \tan \frac { 3 \pi } { 50 } + \ldots + \tan \frac { 19 \pi } { 50 } \tan \frac { 20 \pi } { 50 } = \frac { \tan \frac { 2 \pi } { 5 } } { \tan \frac { \pi } { 50 } } - 20$$

6

1. 1. By applying De Moivre's theorem to \(( \cos \theta + \mathrm { i } \sin \theta ) ^ { 3 }\), show that $$\cos 3 \theta = \cos ^ { 3 } \theta - 3 \cos \theta \sin ^ { 2 } \theta$$
   2. Find a similar expression for \(\sin 3 \theta\).
   3. Deduce that $$\tan 3 \theta = \frac { \tan ^ { 3 } \theta - 3 \tan \theta } { 3 \tan ^ { 2 } \theta - 1 }$$
2. 1. Hence show that \(\tan \frac { \pi } { 12 }\) is a root of the cubic equation $$x ^ { 3 } - 3 x ^ { 2 } - 3 x + 1 = 0$$
   2. Find two other values of \(\theta\), where \(0 < \theta < \pi\), for which \(\tan \theta\) is a root of this cubic equation.
3. Hence show that $$\tan \frac { \pi } { 12 } + \tan \frac { 5 \pi } { 12 } = 4$$