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

5 Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
Hence solve the equation \(4 \cos \theta - \sin \theta = 3\), for \(0 \leqslant \theta \leqslant 2 \pi\).

7

1. By first expanding \(\cos \left( x + 45 ^ { \circ } \right)\), express \(\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places.
2. Hence solve the equation $$\cos \left( x + 45 ^ { \circ } \right) - \sqrt { 2 } \sin x = 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   

4

1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

8

1. Express \(3 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence
   1. solve the equation \(3 \sin \theta + 4 \cos \theta + 1 = 0\), giving all solutions for which \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\),
   2. find the values of the positive constants \(k\) and \(c\) such that $$- 37 \leqslant k ( 3 \sin \theta + 4 \cos \theta ) + c \leqslant 43$$ for all values of \(\theta\).

8

1. Express \(4 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence
   1. solve the equation \(4 \cos \theta - 2 \sin \theta = 3\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
   2. determine the greatest and least values of $$25 - ( 4 \cos \theta - 2 \sin \theta ) ^ { 2 }$$ as \(\theta\) varies, and, in each case, find the smallest positive value of \(\theta\) for which that value occurs.

9

1. Express \(5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence
   1. give details of the transformations needed to transform the curve \(y = 5 \cos \left( \theta - 60 ^ { \circ } \right) + 3 \cos \theta\) to the curve \(y = \sin \theta\),
   2. find the smallest positive value of \(\beta\) satisfying the equation $$5 \cos \left( \frac { 1 } { 3 } \beta - 40 ^ { \circ } \right) + 3 \cos \left( \frac { 1 } { 3 } \beta + 20 ^ { \circ } \right) = 3 .$$ \section*{END OF QUESTION PAPER}

8 The points \(A\) and \(B\) have position vectors relative to the origin \(O\) given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 \sin \alpha \\ 2 \cos \alpha \\ - 1 \end{array} \right) \text { and } \overrightarrow { O B } = \left( \begin{array} { c } 2 \cos \alpha \\ 4 \sin \alpha \\ 3 \end{array} \right)$$ where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). It is given that \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are perpendicular.

1. Calculate the two possible values of \(\alpha\).
2. Calculate the area of triangle \(O A B\) for the smaller value of \(\alpha\) from part (i).

6

1. Express \(\cos \theta + \sqrt { 3 } \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(\alpha\) is acute, expressing \(\alpha\) in terms of \(\pi\).
2. Write down the derivative of \(\tan \theta\). Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { ( \cos \theta + \sqrt { 3 } \sin \theta ) ^ { 2 } } \mathrm {~d} \theta = \frac { \sqrt { 3 } } { 4 }\).

6 In Fig. 6, OAB is a thin bent rod, with \(\mathrm { OA } = a\) metres, \(\mathrm { AB } = b\) metres and angle \(\mathrm { OAB } = 120 ^ { \circ }\). The bent rod lies in a vertical plane. OA makes an angle \(\theta\) above the horizontal. The vertical height BD of B above O is \(h\) metres. The horizontal through A meets BD at C and the vertical through A meets OD at E . \begin{figure}[h] \end{figure}

1. Find angle BAC in terms of \(\theta\). Hence show that $$h = a \sin \theta + b \sin \left( \theta - 60 ^ { \circ } \right) .$$
2. Hence show that \(h = \left( a + \frac { 1 } { 2 } b \right) \sin \theta - \frac { \sqrt { 3 } } { 2 } b \cos \theta\). The rod now rotates about O , so that \(\theta\) varies. You may assume that the formulae for \(h\) in parts (i) and (ii) remain valid.
3. Show that OB is horizontal when \(\tan \theta = \frac { \sqrt { 3 } b } { 2 a + b }\). In the case when \(a = 1\) and \(b = 2 , h = 2 \sin \theta - \sqrt { 3 } \cos \theta\).
4. Express \(2 \sin \theta - \sqrt { 3 } \cos \theta\) in the form \(R \sin ( \theta - \alpha )\). Hence, for this case, write down the maximum value of \(h\) and the corresponding value of \(\theta\).

4 You are given that \(\mathrm { f } ( x ) = \cos x + \lambda \sin x\) where \(\lambda\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving \(R\) and \(\alpha\) in terms of \(\lambda\).
2. Given that the maximum value (as \(x\) varies) of \(\mathrm { f } ( x )\) is 2 , find \(R , \lambda\) and \(\alpha\), giving your answers in exact form.

8

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that
   1. \(\cosh ( \ln a ) \equiv \frac { a ^ { 2 } + 1 } { 2 a }\), where \(a > 0\),
   2. \(\cosh x \cosh y - \sinh x \sinh y \equiv \cosh ( x - y )\).
   3. Use part (i)(b) to show that \(\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1\).
   4. Given that \(R > 0\) and \(a > 1\), find \(R\) and \(a\) such that $$13 \cosh x - 5 \sinh x \equiv R \cosh ( x - \ln a )$$
   5. Hence write down the coordinates of the minimum point on the curve with equation \(y = 13 \cosh x - 5 \sinh x\).

8 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 10 \sin 3 x - 20 \cos 3 x$$ Show that, for large positive \(x\) and independently of the initial conditions, $$y \approx R \sin ( 3 x + \phi )$$ where the constants \(R\) and \(\phi\), such that \(R > 0\) and \(0 < \phi < 2 \pi\), are to be determined correct to 2 decimal places.

9

1. Express \(3 \cos 3 x + 7 \sin 3 x\) in the form \(R \cos ( 3 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. Give full details of a sequence of three transformations needed to transform the curve \(y = \cos x\) to the curve \(y = 3 \cos 3 x + 7 \sin 3 x\).
3. Determine the greatest value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.
4. Determine the least value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.

6. a. Express \(4 \sin x - 5 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\), and give the value of \(\alpha\), in degrees, to 2 decimal places. $$T = \frac { 8400 } { 19 + ( 4 \sin x - 5 \cos x ) ^ { 2 } } , x > 0$$ b. Use your answer to part \(a\) to calculate
i. the minimum value of \(T\).
ii. the smallest value of \(x , x > 0\), at which this minimum value occurs.

9. $$\mathrm { f } ( \theta ) = 4 \cos \theta + 5 \sin \theta \quad \theta \in R$$ a. Express \(\mathrm { f } ( \theta )\) 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. Given that $$\mathrm { g } ( \theta ) = \frac { 135 } { 4 + \mathrm { f } ( \theta ) ^ { 2 } } \quad \theta \in R$$ b.find the range of \(g\).

13.

1. Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places. Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula $$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017. Use Tom's model to
2. find the depth of water at 00:00 hours on 18th October 2017,
3. find the maximum depth of water,
4. find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula $$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$ where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
   By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
5. 1. explain why Jolene's model is not correct,
   2. hence find a suitable model for \(H\) in terms of \(x\).

13.

1. Express \(10 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\) Give the exact value of \(R\) and give the value of \(\alpha\), in degrees, to 2 decimal places. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a75c9ef7-b648-47be-bad1-fc8b315be3df-18_396_1329_388_367} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The height above the ground, \(H\) metres, of a passenger on a Ferris wheel \(t\) minutes after the wheel starts turning, is modelled by the equation $$H = a - 10 \cos ( 80 t ) ^ { \circ } + 3 \sin ( 80 t ) ^ { \circ }$$ where \(a\) is a constant.
   Figure 3 shows the graph of \(H\) against \(t\) for two complete cycles of the wheel.
   Given that the initial height of the passenger above the ground is 1 metre,
2. 1. find a complete equation for the model,
   2. hence find the maximum height of the passenger above the ground.
3. Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle.
   (Solutions based entirely on graphical or numerical methods are not acceptable.) It is decided that, to increase profits, the speed of the wheel is to be increased.
4. How would you adapt the equation of the model to reflect this increase in speed?

10

1. Express \(7 \cos x - 2 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
2. Give details of a sequence of two transformations which maps the curve \(y = \sec x\) onto the curve \(y = \frac { 1 } { 7 \cos x - 2 \sin x }\).