1.05l Double angle formulae: and compound angle formulae

7. (i) Prove that, for \(\cos x \neq 0\), $$\sin 2 x - \tan x \equiv \tan x \cos 2 x$$ (ii) Hence, or otherwise, solve the equation $$\sin 2 x - \tan x = 2 \cos 2 x$$ for \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\).

4. (i) Given that \(\cos x = \sqrt { 3 } - 1\), find the value of \(\cos 2 x\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
(ii) Given that $$2 \cos ( y + 30 ) ^ { \circ } = \sqrt { 3 } \sin ( y - 30 ) ^ { \circ }$$ find the value of \(\tan y\) in the form \(k \sqrt { 3 }\) where \(k\) is a rational constant.

1. (i) Show that

$$\sin ( x + 30 ) ^ { \circ } + \sin ( x - 30 ) ^ { \circ } \equiv a \sin x ^ { \circ }$$ where \(a\) is a constant to be found.
(ii) Hence find the exact value of \(\sin 75 ^ { \circ } + \sin 15 ^ { \circ }\), giving your answer in the form \(b \sqrt { 6 }\).

6. (i) Prove the identity $$2 \cot 2 x + \tan x \equiv \cot x , \quad x \neq \frac { n } { 2 } \pi , \quad n \in \mathbb { Z }$$ (ii) Solve, for \(0 \leq x < \pi\), the equation $$2 \cot 2 x + \tan x = \operatorname { cosec } ^ { 2 } x - 7$$ giving your answers to 2 decimal places.

6. (i) Express \(3 \cos x ^ { \circ } + \sin x ^ { \circ }\) in the form \(R \cos ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(ii) Using your answer to part (a), or otherwise, solve the equation $$6 \cos ^ { 2 } x ^ { \circ } + \sin 2 x ^ { \circ } = 0$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place where appropriate.

3. (i) Use the identity for \(\sin ( A + B )\) to show that $$\sin 3 x \equiv 3 \sin x - 4 \sin ^ { 3 } x$$ (ii) Hence find, in terms of \(\pi\), the solutions of the equation $$\sin 3 x - \sin x = 0$$ for \(x\) in the interval \(0 \leq x < 2 \pi\).

7. (i) Use the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin ^ { 2 } \frac { x } { 2 }$$ (ii) Prove that, for \(\sin x \neq 0\), $$\frac { 1 - \cos x } { \sin x } \equiv \tan \frac { x } { 2 }$$ (iii) Find the values of \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) for which $$\frac { 1 - \cos x } { \sin x } = 2 \sec ^ { 2 } \frac { x } { 2 } - 5$$ giving your answers to 1 decimal place where appropriate.

4. Find the values of \(x\) in the interval \(- 180 < x < 180\) for which $$\tan ( x + 45 ) ^ { \circ } - \tan x ^ { \circ } = 4 ,$$ giving your answers to 1 decimal place.

2 It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac { 12 } { 13 }\). Find the exact value of

1. \(\cot \theta\),
2. \(\cos 2 \theta\).

9

1. Use the identity for \(\cos ( A + B )\) to prove that $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) \equiv \sqrt { 3 } - 2 \sin 2 \theta .$$
2. Hence find the exact value of \(4 \cos 82.5 ^ { \circ } \cos 52.5 ^ { \circ }\).
3. Solve, for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation \(4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = 1\).
4. Given that there are no values of \(\theta\) which satisfy the equation $$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = k ,$$ determine the set of values of the constant \(k\).

7

1. Write down the formula for \(\cos 2 x\) in terms of \(\cos x\).
2. Prove the identity \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } \equiv 4 - 2 \sec ^ { 2 } x\).
3. Solve, for \(0 < x < 2 \pi\), the equation \(\frac { 4 \cos 2 x } { 1 + \cos 2 x } = 3 \tan x - 7\).

5

1. Write down the identity expressing \(\sin 2 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\).
2. Given that \(\sin \alpha = \frac { 1 } { 4 }\) and \(\alpha\) is acute, show that \(\sin 2 \alpha = \frac { 1 } { 8 } \sqrt { 15 }\).
3. Solve, for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\), the equation \(5 \sin 2 \beta \sec \beta = 3\).

9

1. Prove the identity $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta - 3 } { 1 - 3 \tan ^ { 2 } \theta }$$
2. Solve, for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = 4 \sec ^ { 2 } \theta - 3 ,$$ giving your answers correct to the nearest \(0.1 ^ { \circ }\).
3. Show that, for all values of the constant k , the equation $$\tan \left( \theta + 60 ^ { \circ } \right) \tan \left( \theta - 60 ^ { \circ } \right) = \mathrm { K } ^ { 2 }$$ has two roots in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).

5

1. Express \(\tan 2 \alpha\) in terms of \(\tan \alpha\) and hence solve, for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\), the equation $$\tan 2 \alpha \tan \alpha = 8 .$$
2. Given that \(\beta\) is the acute angle such that \(\sin \beta = \frac { 6 } { 7 }\), find the exact value of
   1. \(\operatorname { cosec } \beta\),
   2. \(\cot ^ { 2 } \beta\).

8 The expression \(\mathrm { T } ( \theta )\) is defined for \(\theta\) in degrees by $$\mathrm { T } ( \theta ) = 3 \cos \left( \theta - 60 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) .$$

1. Express \(\mathrm { T } ( \theta )\) in the form \(A \sin \theta + B \cos \theta\), giving the exact values of the constants \(A\) and \(B\). [3]
2. Hence express \(\mathrm { T } ( \theta )\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
3. Find the smallest positive value of \(\theta\) such that \(\mathrm { T } ( \theta ) + 1 = 0\).

2

1. Prove the identity $$\sin \left( x + 30 ^ { \circ } \right) + ( \sqrt { } 3 ) \cos \left( x + 30 ^ { \circ } \right) \equiv 2 \cos x$$ where \(x\) is measured in degrees.
2. Hence express \(\cos 15 ^ { \circ }\) in surd form.

7

1. Write down the formula for \(\tan 2 x\) in terms of \(\tan x\).
2. By letting \(\tan x = t\), show that the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ becomes $$3 t ^ { 4 } - 8 t ^ { 2 } - 3 = 0$$
3. Hence find all the solutions of the equation $$4 \tan 2 x + 3 \cot x \sec ^ { 2 } x = 0$$ which lie in the interval \(0 \leqslant x \leqslant 2 \pi\).

1. (i) Use the identity for \(\cos ( A + B )\) to prove that

$$\cos 2 x \equiv 2 \cos ^ { 2 } x - 1$$ (ii) Prove that, for \(\cos x \neq 0\), $$2 \cos x - \sec x \equiv \sec x \cos 2 x$$ (iii) Hence, or otherwise, find the values of \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\) for which $$2 \cos x - \sec x \equiv 2 \cos 2 x$$

1. (i) Show that the equation

$$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ can be written as $$\sqrt { 3 } \sin x \cos x + \cos ^ { 2 } x = 0$$ (ii) Hence, or otherwise, find in terms of \(\pi\) the solutions of the equation $$2 \sin x + \sec \left( x + \frac { \pi } { 6 } \right) = 0$$ for \(x\) in the interval \(0 \leq x \leq \pi\).

1 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where $$f ( x ) = a + \cos b x , 0 \leqslant x \leqslant 2 \pi ,$$ and \(a\) and \(b\) are positive constants. The curve has stationary points at \(( 0,3 )\) and \(( 2 \pi , 1 )\). \begin{figure}[h] \end{figure}

1. Find \(a\) and \(b\).
2. Find \(\mathrm { f } ^ { - 1 } ( x )\), and state its domain and range.

3

1. Use the formula for \(\sin ( \theta + \phi )\), with \(\theta = 45 ^ { \circ }\) and \(\phi = 60 ^ { \circ }\), to show that \(\sin 105 ^ { \circ } = \frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }\).
2. In triangle ABC , angle \(\mathrm { BAC } = 45 ^ { \circ }\), angle \(\mathrm { ACB } = 30 ^ { \circ }\) and \(\mathrm { AB } = 1\) unit (see Fig. 3). Fig. 3 Using the sine rule, together with the result in part (i), show that \(\mathrm { AC } = \frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }\).

4 The angle \(\theta\) satisfies the equation \(\sin \left( \theta + 45 ^ { \circ } \right) = \cos \theta\).

1. Using the exact values of \(\sin 45 ^ { \circ }\) and \(\cos 45 ^ { \circ }\), show that \(\tan \theta = \sqrt { 2 } - 1\).
2. Find the values of \(\theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

6 Solve the equation \(\tan \left( \theta + 45 ^ { \circ } \right) = 1 - 2 \tan \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 90 ^ { \circ }\). Section B (36 marks)

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

8 The height of tide at the entrance to a harbour on a particular day may be modelled by the function \(h = 3 + 2 \sin 30 t + 1.5 \cos 30 t\) where \(h\) is measured in metres, \(t\) in hours after midnight and \(30 t\) is in degrees.
[0pt] [The values 2 and 1.5 represent the relative effects of the moon and sun respectively.]

1. Show that \(2 \sin 30 t + 1.5 \cos 30 t\) can be written in the form \(2.5 \sin ( 30 t + \alpha )\), where \(\alpha\) is to be determined.
2. Find the height of tide at high water and the first time that this occurs after midnight.
3. Find the range of tide during the day.
4. Sketch the graph of \(h\) against \(t\) for \(0 \leq t \leq 12\), indicating the maximum and minimum points.
5. A sailing boat may enter the harbour only if there is at least 2 metres of water. Find the times during this morning when it may enter the harbour.
6. From your graph estimate the time at which the water falling fastest and the rate at which it is falling.