1.05l Double angle formulae: and compound angle formulae

2 Let \(u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }\).

1. Using the formulae for \(\cos P \pm \cos Q\) given in the List of Formulae MF10, show that $$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$
2. Use the method of differences to find \(\sum _ { n = 1 } ^ { N } u _ { n }\).
3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.

7 Use de Moivre's theorem to express \(\sin ^ { 6 } \theta\) in the form $$a + b \cos 2 \theta + c \cos 4 \theta + d \cos 6 \theta$$ where \(a , b , c , d\) are constants to be found. Hence evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } 2 x d x$$ leaving your answer in terms of \(\pi\).

5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 n \theta } { \cos \theta } \mathrm {~d} \theta\), where \(n\) is a non-negative integer.

1. Use the identity \(\sin P + \sin Q \equiv 2 \sin \frac { 1 } { 2 } ( P + Q ) \cos \frac { 1 } { 2 } ( P - Q )\) to show that $$I _ { n } + I _ { n - 1 } = \frac { 2 } { 2 n - 1 } , \text { for all positive integers } n$$
2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 8 \theta } { \cos \theta } d \theta\).

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

7

1. Show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$\text { into } \quad \begin{aligned} x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y & = 3 + 20 \sin ( \ln x ) \\ \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 6 y & = 3 + 20 \sin t \end{aligned}$$ (7 marks)
2. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 6 y = 3 + 20 \sin t$$ (11 marks)
3. Write down the general solution of the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = 3 + 20 \sin ( \ln x )$$

12 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{1ba9fa5f-310f-4429-9bd1-4004852d5b3e-6_716_479_292_794} The diagram shows the curve \(y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }\), for \(x \geqslant 0\), and the normal to the curve at the point \(\left( \frac { 1 } { 4 } \pi , 0 \right)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }\).
[0pt] [10]

10

1. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-7_599_780_267_328} The diagram shows triangle \(A B C\). The perpendicular from \(C\) to \(A B\) meets \(A B\) at \(D\). Angle \(A C D = x\), angle \(D C B = y\), length \(B C = a\) and length \(A C = b\).
   1. Explain why the length of \(C D\) can be written as \(a \cos y\).
   2. Show that the area of the triangle \(A D C\) is given by \(\frac { 1 } { 2 } a b \sin x \cos y\).
   3. Hence, or otherwise, show that \(\sin ( x + y ) = \sin x \cos y + \cos x \sin y\).
2. Given that \(\sin \left( 30 ^ { \circ } + \alpha \right) = \cos \left( 45 ^ { \circ } - \alpha \right)\), show that \(\tan \alpha = 2 + \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 }\).

7

1. Use the result \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\) to show that \(\cos ( A - B ) = \cos A \cos B + \sin A \sin B\). The function \(\mathrm { f } ( \theta )\) is defined as \(\cos \left( \theta + 30 ^ { \circ } \right) \cos \left( \theta - 30 ^ { \circ } \right)\), where \(\theta\) is in degrees.
2. Show that \(f ( \theta ) = \cos ^ { 2 } \theta - \frac { 1 } { 4 }\).
3. (i) Determine the following.

4 Prove that \(\sin ^ { 2 } ( \theta + 45 ) ^ { \circ } - \cos ^ { 2 } ( \theta + 45 ) ^ { \circ } \equiv \sin 2 \theta ^ { \circ }\).

6 It is given that the angle \(\theta\) satisfies the equation \(\sin \left( 2 \theta + \frac { 1 } { 4 } \pi \right) = 3 \cos \left( 2 \theta + \frac { 1 } { 4 } \pi \right)\).

1. Show that \(\tan 2 \theta = \frac { 1 } { 2 }\).
2. Hence find, in surd form, the exact value of \(\tan \theta\), given that \(\theta\) is an obtuse angle.

1. (a) Show that the equation

$$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.)

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\frac { 1 } { \cos \theta } + \tan \theta \equiv \frac { \cos \theta } { 1 - \sin \theta } \quad \theta \neq ( 2 n + 1 ) 90 ^ { \circ } \quad n \in \mathbb { Z }$$ Given that \(\cos 2 x \neq 0\)
2. solve for \(0 < x < 90 ^ { \circ }\) $$\frac { 1 } { \cos 2 x } + \tan 2 x = 3 \cos 2 x$$ giving your answers to one decimal place.

14. Given that $$2 \cos ( x + 60 ) ^ { 0 } = \sin ( x - 30 ) ^ { 0 }$$ a. Show, without using a calculator, that $$\tan x = \frac { \sqrt { 3 } } { 3 }$$ b. Hence solve, for \(0 \leq \theta < 360 ^ { 0 }\) $$2 \cos ( 2 \theta + 90 ) ^ { 0 } = \sin ( 2 \theta ) ^ { 0 }$$

10. a. Show that $$\sin 3 A \equiv 3 \sin A - 4 \sin ^ { 3 } A$$ b. Hence solve, for \(- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }\) the equation $$1 + \sin 3 \theta = \cos ^ { 2 } \theta$$

12. \begin{figure}[h] \end{figure} Figure 5 shows part of the curve \(C\) with parametric equations $$x = 2 \cos \theta \quad y = \sin 2 \theta \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The region \(R\), shown shaded in figure 5, is bounded by the curve \(C\), the line \(x = \sqrt { 2 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid revolution.
a. Show that the volume of the solid of revolution formed is given by the integral. $$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \sin ^ { 3 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$ where \(k\) is a constant. \includegraphics[max width=\textwidth, alt={}, center]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-22_164_1148_54_118}
b. Hence, find the exact value for this volume, giving your answer in the form \(p \pi \sqrt { 2 }\) where \(p\) is a constant.

12. a. Prove that $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } \equiv \sin ^ { 2 } x$$ b. Hence solve, for \(- 360 ^ { \circ } < x < 360 ^ { \circ }\), the equation $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } = \frac { \cos 2 x } { 2 }$$

1. Given that

$$y = \frac { 3 \sin \theta } { 2 \sin \theta + 2 \cos \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { d y } { d \theta } = \frac { A } { 1 + \sin 2 \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(A\) is a rational constant to be found.

1. (a) Solve, for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), the equation

$$5 \sin 2 \theta = 9 \tan \theta$$ giving your answers, where necessary, to one decimal place.
[0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
(b) Deduce the smallest positive solution to the equation $$5 \sin \left( 2 x - 50 ^ { \circ } \right) = 9 \tan \left( x - 25 ^ { \circ } \right)$$

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

Solutions relying entirely on calculator technology are not acceptable.

1. Given that $$2 \sin \left( x - 60 ^ { \circ } \right) = \cos \left( x - 30 ^ { \circ } \right)$$ show that $$\tan x = 3 \sqrt { 3 }$$
2. Hence or otherwise solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$2 \sin 2 \theta = \cos \left( 2 \theta + 30 ^ { \circ } \right)$$ giving your answers to one decimal place.

1. (a) Given that \(a\) is a positive constant, use the substitution \(x = a \sin ^ { 2 } \theta\) to show that

$$\int _ { 0 } ^ { a } x ^ { \frac { 1 } { 2 } } \sqrt { a - x } \mathrm {~d} x = \frac { 1 } { 2 } a ^ { 2 } \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 2 } 2 \theta \mathrm {~d} \theta$$ (b) Hence use algebraic integration to show that $$\int _ { 0 } ^ { a } x ^ { \frac { 1 } { 2 } } \sqrt { a - x } d x = k \pi a ^ { 2 }$$ where \(k\) is a constant to be found.

9. $$f ( x ) = \frac { 50 x ^ { 2 } + 38 x + 9 } { ( 5 x + 2 ) ^ { 2 } ( 1 - 2 x ) } \quad x \neq - \frac { 2 } { 5 } \quad x \neq \frac { 1 } { 2 }$$ Given that \(\mathrm { f } ( x )\) can be expressed in the form $$\frac { A } { 5 x + 2 } + \frac { B } { ( 5 x + 2 ) ^ { 2 } } + \frac { C } { 1 - 2 x }$$ where \(A\), \(B\) and \(C\) are constants

1. 1. find the value of \(B\) and the value of \(C\)
   2. show that \(A = 0\)
2. 1. Use binomial expansions to show that, in ascending powers of \(x\) $$f ( x ) = p + q x + r x ^ { 2 } + \ldots$$ where \(p , q\) and \(r\) are simplified fractions to be found.
   2. Find the range of values of \(x\) for which this expansion is valid.

14. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) , y = 2 \sin t , \quad 0 < t \leqslant 2 \pi$$ Show that a Cartesian equation of \(C\) can be written in the form $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where \(a\) and \(b\) are integers to be found.

1. (a) Prove that

$$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.

1. (a) Prove

$$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } \equiv 2 \cot 2 \theta \quad \theta \neq ( 90 n ) ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence solve, for \(90 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\frac { \cos 3 \theta } { \sin \theta } + \frac { \sin 3 \theta } { \cos \theta } = 4$$ giving any solutions to one decimal place.

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 $$2 \tan \theta \left( 8 \cos \theta + 23 \sin ^ { 2 } \theta \right) = 8 \sin 2 \theta \left( 1 + \tan ^ { 2 } \theta \right)$$ may be written as $$\sin 2 \theta \left( A \cos ^ { 2 } \theta + B \cos \theta + C \right) = 0$$ where \(A , B\) and \(C\) are constants to be found.
2. Hence, solve for \(360 ^ { \circ } \leqslant x \leqslant 540 ^ { \circ }\) $$2 \tan x \left( 8 \cos x + 23 \sin ^ { 2 } x \right) = 8 \sin 2 x \left( 1 + \tan ^ { 2 } x \right) \quad x \in \mathbb { R } \quad x \neq 450 ^ { \circ }$$