1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc

14 Douglas wants to construct a model for the height of the tide in Liverpool during the day, using a cosine graph to represent the way the height changes. He knows that the first high tide of the day measures 8.55 m and the first low tide of the day measures 1.75 m . Douglas uses \(t\) for time and \(h\) for the height of the tide in metres. With his graph-drawing software set to degrees, he begins by drawing the graph of \(\mathrm { h } = 5.15 + 3.4\) cost.

1. Verify that this equation gives the correct values of \(h\) for the high and low tide. Douglas also knows that the first high tide of the day occurs at 1 am and the first low tide occurs at 7.20 am. He wants \(t\) to represent the time in hours after midnight, so he modifies his equation to \(h = 5.15 + 3.4 \cos ( a t + b )\).
2. 1. Show that Douglas's modified equation gives the first high tide of the day occurring at the correct time if \(\mathrm { a } + \mathrm { b } = 0\).
   2. Use the time of the first low tide of the day to form a second equation relating \(a\) and \(b\).
   3. Hence show that \(a = 28.42\) correct to 2 decimal places.
3. Douglas can only sail his boat when the height of the tide is at least 3 m . Use the model to predict the range of times that morning when he cannot sail.
4. The next high tide occurs at 12.59 pm when the height of the tide is 8.91 m . Comment on the suitability of Douglas's model.

6

1. Express \(7 \cos x - 24 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(0 < \alpha < \frac { \pi } { 2 }\).
2. Write down the range of the function $$f ( x ) = 12 + 7 \cos x - 24 \sin x , \quad 0 \leqslant x \leqslant 2 \pi .$$

8 In this question you must show detailed reasoning.

1. Express \(\cos x + \sqrt { 3 } \sin x\) in the form \(\mathrm { R } \sin ( \mathrm { x } + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the values of \(R\) and \(\alpha\) in exact form.
2. Hence solve the equation \(\cos x = \sqrt { 3 } ( 1 - \sin x )\) for values of \(x\) in the interval \(- \pi \leqslant x \leqslant \pi\). Give the roots of this equation in exact form.

9

1. Express \(\cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\) and \(R\) is positive and given in exact form. The function \(\mathrm { f } ( \theta )\) is defined by \(\mathrm { f } ( \theta ) = \frac { 1 } { ( k + \cos \theta + 2 \sin \theta ) } , 0 \leq \theta \leq 2 \pi , k\) is a constant.
2. The maximum value of \(\mathrm { f } ( \theta )\) is \(\frac { ( 3 + \sqrt { 5 } ) } { 4 }\). Find the value of \(k\).

5. (a) Express \(3 \cos x ^ { \circ } + \sin x ^ { \circ }\) in the form \(R \cos ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(b) 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.

7. (a) Express \(4 \sin x + 3 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(b) State the minimum value of \(4 \sin x + 3 \cos x\) and the smallest positive value of \(x\) for which this minimum value occurs.
(c) Solve the equation $$4 \sin 2 \theta + 3 \cos 2 \theta = 2$$ for \(\theta\) in the interval \(0 \leq \theta \leq \pi\), giving your answers to 2 decimal places.

5. (a) Express \(\sqrt { 3 } \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(b) State the maximum value of \(\sqrt { 3 } \sin \theta + \cos \theta\) and the smallest positive value of \(\theta\) for which this maximum value occurs.
(c) Solve the equation $$\sqrt { 3 } \sin \theta + \cos \theta + \sqrt { 3 } = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\), giving your answers in terms of \(\pi\).

4. (a) Express \(2 \sin x ^ { \circ } - 3 \cos x ^ { \circ }\) in the form \(R \sin ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(b) Show that the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ can be written in the form $$2 \sin x ^ { \circ } - 3 \cos x ^ { \circ } = 1 .$$ (c) Solve the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2 ,$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place.

8. $$f ( x ) = 2 x + \sin x - 3 \cos x$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [0.7, 0.8].
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where it crosses the \(y\)-axis.
3. Find the values of the constants \(a , b\) and \(c\), where \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\), such that $$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
4. Hence find the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\), giving your answers to 2 decimal places.

1

1. Express \(2 \sin x + 5 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. 1. Write down the maximum value of \(2 \sin x + 5 \cos x\).
   2. Find the value of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) at which this maximum occurs, giving the value of \(x\) to the nearest \(0.1 ^ { \circ }\).

3

1. 1. Express \(3 \cos x + 2 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
      (3 marks)
   2. Hence find the minimum value of \(3 \cos x + 2 \sin x\) and the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) where the minimum occurs. Give your value of \(x\) to the nearest \(0.1 ^ { \circ }\).
2. 1. Show that \(\cot x - \sin 2 x = \cot x \cos 2 x\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
   2. Hence, or otherwise, solve the equation $$\cot x - \sin 2 x = 0$$ in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).

2

1. Express \(\sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Hence find the values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) for which $$\sin x - 3 \cos x + 2 = 0$$ giving your values of \(x\) to the nearest degree.

5

1. 1. Express \(3 \sin x + 4 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
   2. Hence solve the equation \(3 \sin 2 \theta + 4 \cos 2 \theta = 5\) in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your solutions to the nearest \(0.1 ^ { \circ }\).
2. 1. Show that the equation \(\tan 2 \theta \tan \theta = 2\) can be written as \(2 \tan ^ { 2 } \theta = 1\).
   2. Hence solve the equation \(\tan 2 \theta \tan \theta = 2\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), giving your solutions to the nearest \(0.1 ^ { \circ }\).
3. 1. Use the Factor Theorem to show that \(2 x - 1\) is a factor of \(8 x ^ { 3 } - 4 x + 1\).
   2. Show that \(4 \cos 2 \theta \cos \theta + 1\) can be written as \(8 x ^ { 3 } - 4 x + 1\) where \(x = \cos \theta\).
   3. Given that \(\theta = 72 ^ { \circ }\) is a solution of \(4 \cos 2 \theta \cos \theta + 1 = 0\), use the results from parts (c)(i) and (c)(ii) to show that the exact value of \(\cos 72 ^ { \circ }\) is \(\frac { ( \sqrt { 5 } - 1 ) } { p }\) where \(p\) is an integer.
      [0pt] [3 marks]

2

1. Express \(2 \cos x - 5 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\), giving your value of \(\alpha\), in radians, to three significant figures.
2. 1. Hence find the value of \(x\) in the interval \(0 < x < 2 \pi\) for which \(2 \cos x - 5 \sin x\) has its maximum value. Give your value of \(x\) to three significant figures.
   2. Use your answer to part (a) to solve the equation \(2 \cos x - 5 \sin x + 1 = 0\) in the interval \(0 < x < 2 \pi\), giving your solutions to three significant figures.
      [0pt] [3 marks]

2. A particle \(P\) moves along the \(x\)-axis such that its position \(x\) metres, after \(t\) seconds, is given by $$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$

1. 1. Show that the motion of the particle \(P\) is Simple Harmonic. State the value of \(x\) at the centre of motion.
   2. Show that the period of the motion of \(P\) is 2 s and determine the amplitude. Suppose that another particle \(Q\) is introduced so that it also moves along the \(x\)-axis with Simple Harmonic Motion with centre of motion, \(O\), and period equal to that of particle \(P\). When \(t = 0\), the particle \(Q\) is at \(O\) and when it is \(2 \sqrt { 3 } \mathrm {~m}\) from \(O\) its speed is \(2 \pi \mathrm {~ms} ^ { - 1 }\).
2. Find the amplitude of particle \(Q\).
3. Determine the time when particles \(P\) and \(Q\) first meet.

1. On a particular day, the depth of water in a river estuary at a specific location is modelled by the equation

$$D = 2 \sin \left( \frac { x } { 3 } \right) + 3 \cos \left( \frac { x } { 3 } \right) + 6 \quad 0 \leqslant x \leqslant 7 \pi$$ where the depth of water is \(D\) metres at time \(x\) hours after midnight on that day.

1. Write down the depth of water at midnight, according to the model. Using the substitution \(t = \tan \left( \frac { x } { 6 } \right)\)
2. show that equation (I) can be re-written as $$D = \frac { 3 t ^ { 2 } + 4 t + 9 } { 1 + t ^ { 2 } }$$
3. Hence determine, according to the model, the time after midnight when the depth of water is 5 metres for the first time. Give your answer to the nearest minute.

1. During 2029, the number of hours of daylight per day in London, H, is modelled by the equation

$$H = 0.3 \sin \left( \frac { x } { 60 } \right) - 4 \cos \left( \frac { x } { 60 } \right) + 11.5 \quad 0 \leqslant x < 365$$ where \(x\) is the number of days after 1st January 2029 and the angle is in radians.

1. Show that, according to the model, the number of hours of daylight in London on the 31st January 2029 will be 8.13 to 3 significant figures.
2. Use the substitution \(t = \tan \left( \frac { x } { 120 } \right)\) to show that \(H\) can be written as $$H = \frac { a t ^ { 2 } + b t + c } { 1 + t ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are constants to be determined.
3. Hence determine, according to the model, the date of the first day of 2029 when there will be at least 12 hours of daylight in London.

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 \(0 ^ { \circ } \leqslant A < 180 ^ { \circ }\).

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.

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 \(0 ^ { \circ } < x < 360 ^ { \circ }\).

2

1. Express \(\sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) in radians to two decimal places.
2. Hence:
   1. write down the minimum value of \(\sin x - 3 \cos x\);
   2. find the value of \(x\) in the interval \(0 < x < 2 \pi\) at which this minimum value occurs, giving your value of \(x\) in radians to two decimal places.

2

1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\), in radians, to three decimal places.
2. 1. Hence write down the minimum value of \(\cos x + 3 \sin x\).
   2. Find the value of \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\) at which this minimum occurs, giving your answer, in radians, to three decimal places.
3. Solve the equation \(\cos x + 3 \sin x = 2\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving all solutions, in radians, to three decimal places.

1

1. Express \(2 \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R\) is a positive constant and \(\alpha\) is an acute angle. Give your value of \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Solve the equation \(2 \sin x + \cos x = 1\) for \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).