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

3

1. Express \(4 \cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 360 ^ { \circ }\), giving your value for \(\alpha\) to the nearest \(0.1 ^ { \circ }\).
2. Hence solve the equation \(4 \cos x + 3 \sin x = 2\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\), giving all solutions to the nearest \(0.1 ^ { \circ }\).
3. Write down the minimum value of \(4 \cos x + 3 \sin x\) and find the value of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\) at which this minimum value occurs.

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.

1 Fig. 1 shows part of the graph of \(y = \sin x - \sqrt { 3 } \cos x\). \begin{figure}[h] \end{figure} Express \(\sin x - \sqrt { 3 } \cos x\) 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 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 \(0 < x < 2 \pi\).

8 Mike, an amateur astronomer who lives in the South of England, wants to know how the number of hours of darkness changes through the year. On various days between February and September he records the length of time, \(H\) hours, of darkness along with \(t\), the number of days after 1 January. His results are shown in Figure 1 below. \begin{figure}[h] \end{figure} Mike models this data using the equation $$H = 3.87 \sin \left( \frac { 2 \pi ( t + 101.75 ) } { 365 } \right) + 11.7$$ 8

1. Find the minimum number of hours of darkness predicted by Mike's model. Give your answer to the nearest minute.
   [0pt] [2 marks] 8
2. Find the maximum number of consecutive days where the number of hours of darkness predicted by Mike's model exceeds 14
   8
3. Mike's friend Sofia, who lives in Spain, also records the number of hours of darkness on various days throughout the year. Her results are shown in Figure 2 below. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-10_933_1537_561_248}
   \end{figure} Sofia attempts to model her data by refining Mike's model.
   She decides to increase the 3.87 value, leaving everything else unchanged.
   Explain whether Sofia's refinement is appropriate. \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-11_2488_1730_219_141} \(9 \quad\) Chloe is attempting to write \(\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } }\) as partial fractions, with constant numerators. Her incorrect attempt is shown below. Step 1 $$\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } } \equiv \frac { A } { x + 1 } + \frac { B } { ( x + 2 ) ^ { 2 } }$$ Step 2 $$2 x ^ { 2 } + x \equiv A ( x + 2 ) ^ { 2 } + B ( x + 1 )$$ Step 3 $$\begin{aligned} & \text { Let } x = - 1 \Rightarrow A = 1 \\ & \text { Let } x = - 2 \Rightarrow B = - 6 \end{aligned}$$ Answer $$\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } } \equiv \frac { 1 } { x + 1 } - \frac { 6 } { ( x + 2 ) ^ { 2 } }$$

9 A robotic arm which is attached to a flat surface at the origin \(O\), is used to draw a graphic design. The arm is made from two rods \(O P\) and \(P Q\), each of length \(d\), which are joined at \(P\).
A pen is attached to the arm at \(Q\).
The coordinates of the pen are controlled by adjusting the angle \(O P Q\) and the angle \(\theta\) between \(O P\) and the \(x\)-axis. For this particular design the pen is made to move so that the two angles are always equal to each other with \(0 \leq \theta \leq \frac { \pi } { 2 }\) as shown in Figure 2. \begin{figure}[h] \end{figure} 9

1. Show that the \(x\)-coordinate of the pen can be modelled by the equation $$x = d \left( \cos \theta + \sin \left( 2 \theta - \frac { \pi } { 2 } \right) \right)$$ 9
2. Hence, show that $$x = d \left( 1 + \cos \theta - 2 \cos ^ { 2 } \theta \right)$$ 9
3. It can be shown that $$x = \frac { 9 d } { 8 } - d \left( \cos \theta - \frac { 1 } { 4 } \right) ^ { 2 }$$ State the greatest possible value of \(x\) and the corresponding value of \(\cos \theta\) 9
4. Figure 3 below shows the arm when the \(x\)-coordinate is at its greatest possible value. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-14_570_773_456_630}
   \end{figure} Find, in terms of \(d\), the exact distance \(O Q\). \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-15_2488_1716_219_153}

12 Calculate the maximum and minimum values of \(\frac { 1 } { 2 + \cos \theta + \sqrt { 2 } \sin \theta }\) and the smallest positive values of \(\theta\) for which they occur.

9

1. Show that \(\sin \theta + \sqrt { 3 } \cos \theta\) can be expressed in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the values of \(R\) and \(\alpha\).
2. Hence find the value of \(\theta\), where \(0 < \theta < \pi\), such that \(\sin \theta + \sqrt { 3 } \cos \theta = 0.8\).

8

1. Express \(\sin x - \sqrt { 8 } \cos x\) in the form \(R \sin ( x - \alpha )\) where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\).
2. Hence write down the maximum value of \(\sin x - \sqrt { 8 } \cos x\) and find the smallest positive value of \(x\) for which it occurs.

9 In this question, \(x\) denotes an angle measured in degrees.

1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).

The \(n\)th term of a number sequence is denoted by \(x _ { n }\). The \(( n + 1 )\) th term is defined by \(x _ { n + 1 } = 4 x _ { n } - 3\) and \(x _ { 3 } = 113\). a) Find the values of \(x _ { 2 }\) and \(x _ { 1 }\).
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.
a) Express \(5 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find the minimum value of \(\frac { 4 } { 5 \sin x - 12 \cos x + 15 }\).
c) Solve the equation $$5 \sin x - 12 \cos x + 3 = 0$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). a) Find the range of values of \(x\) for which \(| 1 - 3 x | > 7\).
b) Sketch the graph of \(y = | 1 - 3 x | - 7\). Clearly label the minimum point and the points where the graph crosses the \(x\)-axis.

a) Express \(9 \cos x + 40 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). b) Find the maximum possible value of \(\frac { 12 } { 9 \cos x + 40 \sin x + 47 }\).

\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sin 2x + 3\cos 2x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3.

1. Find an expression for \(\frac{dy}{dx}\). [2]
2. By first expressing \(\frac{dy}{dx}\) in the form \(R\cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 significant figures. [8]

1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
2. Find the maximum and minimum values of \(2 \cos \theta + 5 \sin \theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. [2]

The temperature \(T °C\), of an unheated building is modelled using the equation $$T = 15 + 2\cos\left(\frac{\pi t}{12}\right) + 5\sin\left(\frac{\pi t}{12}\right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.

1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
2. Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 °C\). [6]