Harmonic Form

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. Express \(3 \cos \theta + 2 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), stating the exact value of \(R\) and giving the value of \(\alpha\) correct to 1 decimal place.
2. Solve the equation $$3 \cos \theta + 2 \sin \theta = 3.5$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
3. The graph of \(y = 3 \cos \theta + 2 \sin \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), has one stationary point. State the coordinates of this point. \includegraphics[max width=\textwidth, alt={}, center]{9b103197-7ba0-427a-b983-34edb51b6cca-3_421_823_299_662} The diagram shows the curve \(y = 2 x \mathrm { e } ^ { - x }\) and its maximum point \(P\). Each of the two points \(Q\) and \(R\) on the curve has \(y\)-coordinate equal to \(\frac { 1 } { 2 }\).

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.

8. (a) Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence find the maximum value of \(3 \cos \theta + 4 \sin \theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. The temperature, \(\mathrm { f } ( t )\), of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
(c) Calculate the minimum temperature of the warehouse as given by this model.
(d) Find the value of \(t\) when this minimum temperature occurs.

7. (a) Express \(\cos x + 4 \sin x\) in the form \(R \cos ( x - \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. A scientist is studying the behaviour of seabirds in a colony. She models the height above sea level, \(H\) metres, of one of the birds in the colony by the equation $$H = \frac { 24 } { 3 + \cos \left( \frac { 1 } { 2 } t \right) + 4 \sin \left( \frac { 1 } { 2 } t \right) } \quad 0 \leqslant t \leqslant 6.5$$ where \(t\) seconds is the time after it leaves the nest. Find, according to the model,
(b) the minimum height of the seabird above sea level, giving your answer to the nearest cm,
(c) the value of \(t\), to 2 decimal places, when \(H = 10\) \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-21_2255_50_314_34}

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.

5 Express \(\sqrt { 3 } \sin x - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Express \(\alpha\) in the form \(k \pi\). Find the exact coordinates of the maximum point of the curve \(y = \sqrt { 3 } \sin x - \cos x\) for which \(0 < x < 2 \pi\).

7 Fig. 1 shows part of the graph of \(y = \sin x \quad \sqrt { 3 } \cos x\). \begin{figure}[h] \end{figure} Express \(\quad \sqrt { } \quad\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha \leqslant \frac { 1 } { 2 } \pi\).
Hence write down the exact coordinates of the turning point P .

7

1. Express \(5 \cos \theta - 12 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation \(5 \cos \theta - 12 \sin \theta = 8\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
3. Find the greatest possible value of $$7 + 5 \cos \frac { 1 } { 2 } \phi - 12 \sin \frac { 1 } { 2 } \phi$$ as \(\phi\) varies, and determine the smallest positive value of \(\phi\) for which this greatest value occurs.
   [0pt] [4]

1 Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\).

1 Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\).

1. (a) Express \(2 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.

The first three terms of an arithmetic sequence are $$\cos x \quad \cos x + \sin x \quad \cos x + 2 \sin x \quad x \neq n \pi$$ Given that \(S _ { 9 }\) represents the sum of the first 9 terms of this sequence as \(x\) varies,
(b) (i) find the exact maximum value of \(S _ { 9 }\) (ii) deduce the smallest positive value of \(x\) at which this maximum value of \(S _ { 9 }\) occurs.

6

1. Show that the equation ( \(\sqrt { } 2\) ) \(\operatorname { cosec } x + \cot x = \sqrt { } 3\) can be expressed in the form \(R \sin ( x - \alpha ) = \sqrt { } 2\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(( \sqrt { } 2 ) \operatorname { cosec } x + \cot x = \sqrt { } 3\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

4

1. Express \(( \sqrt { } 3 ) \cos x + \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact values of \(R\) and \(\alpha\).
2. Hence show that $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { ( ( \sqrt { } 3 ) \cos x + \sin x ) ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \sqrt { } 3$$

8

1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt { 3 } \sin x - 3 \cos x + 4\) Fully justify your answer.
   8
2. (ii) Find the greatest value of \(\frac { 1 } { \sqrt { 3 } \sin x - 3 \cos x + 4 }\)

3 A curve has equation \(y = 2 \sin 2 x - 5 \cos 2 x + 6\) and is defined for \(0 \leqslant x \leqslant \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures.

$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$ Given that \(\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),

1. find the value of \(R\) and the value of \(\alpha\).
2. Hence solve the equation $$7 \cos 2 x - 24 \sin 2 x = 12.5$$ for \(0 \leqslant x < 180 ^ { \circ }\), giving your answers to 1 decimal place.
3. Express \(14 \cos ^ { 2 } x - 48 \sin x \cos x\) in the form \(a \cos 2 x + b \sin 2 x + c\), where \(a , b\), and \(c\) are constants to be found.
4. Hence, using your answers to parts (a) and (c), deduce the maximum value of $$14 \cos ^ { 2 } x - 48 \sin x \cos x$$

2 Given that $$6 \cos \theta + 8 \sin \theta \equiv R \cos ( \theta + \alpha )$$ find the value of \(R\). Circle your answer. 681014

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 $$f ( x ) = 12 + 7 \cos x - 24 \sin x , \quad 0 \leqslant x \leqslant 2 \pi .$$

7 Two quantities, \(x\) and \(\theta\), vary with time and are related by the equation \(x = 5 \sin \theta - 4 \cos \theta\).

1. Find the value of \(x\) when \(\theta = \frac { \pi } { 2 }\).
2. When \(\theta = \frac { \pi } { 2 }\), its rate of increase (in suitable units) is given by \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = 0.1\). Show that at that moment \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4\).

1. A curve \(C\) has equation

$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$ The point \(A ( 0,4 )\) lies on \(C\).

1. Find an equation of the normal to the curve \(C\) at \(A\).
2. Express \(y\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 significant figures.
3. Find the coordinates of the points of intersection of the curve \(C\) with the \(x\)-axis. Give your answers to 2 decimal places.

8

1. By first expanding \(2 \sin \left( x - 30 ^ { \circ } \right)\), express \(2 \sin \left( x - 30 ^ { \circ } \right) - \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [5]
2. Hence solve the equation $$2 \sin \left( x - 30 ^ { \circ } \right) - \cos x = 1$$ for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

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 \(k\).

1. (a) Express \(2 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 2 decimal places.
   (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\),

$$\frac { 2 } { 2 \cos \theta - \sin \theta - 1 } = 15$$ Give your answers to one decimal place.
(c) Use your solutions to parts (a) and (b) to deduce the smallest positive value of \(\theta\) for which $$\frac { 2 } { 2 \cos \theta + \sin \theta - 1 } = 15$$ Give your answer to one decimal place.

2

1. Express \(5 \sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
2. Hence state the greatest and least possible values of \(( 5 \sin x - 3 \cos x ) ^ { 2 }\).

5 It is given that \(a = \sin \theta - 3 \cos \theta\) and \(b = 3 \sin \theta + \cos \theta\), where \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).

1. Show that \(a ^ { 2 } + b ^ { 2 }\) has a constant value for all values of \(\theta\).
2. Find the values of \(\theta\) for which \(2 a = b\).