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

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.

3. (a) Express \(5 \cos x - 3 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(b) Hence, or otherwise, solve the equation $$5 \cos x - 3 \sin x = 4$$ for \(0 \leqslant x < 2 \pi\), giving your answers to 2 decimal places.

1. (a) Express \(7 \cos x - 24 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 decimal places.
   (b) Hence write down the minimum value of \(7 \cos x - 24 \sin x\).
   (c) Solve, for \(0 \leqslant x < 2 \pi\), the equation

$$7 \cos x - 24 \sin x = 10$$ giving your answers to 2 decimal places.

1. (a) Express \(6 \cos \theta + 8 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).

Give the value of \(\alpha\) to 3 decimal places.
(b) $$\mathrm { p } ( \theta ) = \frac { 4 } { 12 + 6 \cos \theta + 8 \sin \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Calculate

1. the maximum value of \(\mathrm { p } ( \theta )\),
2. the value of \(\theta\) at which the maximum occurs.

5. \begin{figure}[h] \end{figure} The curve shown in Figure 1 has equation $$x = 3 \sin y + 3 \cos y , \quad - \frac { \pi } { 4 } < y < \frac { \pi } { 4 }$$

1. Express the equation of the curve in the form \(x = R \sin ( y + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
2. Find the coordinates of the point on the curve where the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) is \(\frac { 1 } { 2 }\). Give your answers to 3 decimal places.

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 \(0 \leqslant \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.

1. (a) Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
   (b) Hence find the greatest value of \(( 3 \sin x + 2 \cos x ) ^ { 4 }\).
   (c) Solve, for \(0 < x < 2 \pi\), the equation

$$3 \sin x + 2 \cos x = 1$$ giving your answers to 3 decimal places.

2. $$f ( x ) = 5 \cos x + 12 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\),

1. find the value of \(R\) and the value of \(\alpha\) to 3 decimal places.
2. Hence solve the equation $$5 \cos x + 12 \sin x = 6$$ for \(0 \leqslant x < 2 \pi\).
3. 1. Write down the maximum value of \(5 \cos x + 12 \sin x\).
   2. Find the smallest positive value of \(x\) for which this maximum value occurs.

1. (a) Use the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\), to show that

$$\cos 2 A = 1 - 2 \sin ^ { 2 } A$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 3 \sin 2 x \\ & C _ { 2 } : \quad y = 4 \sin ^ { 2 } x - 2 \cos 2 x \end{aligned}$$ (b) Show that the \(x\)-coordinates of the points where \(C _ { 1 }\) and \(C _ { 2 }\) intersect satisfy the equation $$4 \cos 2 x + 3 \sin 2 x = 2$$ (c) Express \(4 \cos 2 x + 3 \sin 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places.
(d) Hence find, for \(0 \leqslant x < 180 ^ { \circ }\), all the solutions of $$4 \cos 2 x + 3 \sin 2 x = 2$$ giving your answers to 1 decimal place.

7. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 4 decimal places.
(b) (i) Find the maximum value of \(2 \sin \theta - 1.5 \cos \theta\).
(ii) Find the value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum occurs. Tom models the height of sea water, \(H\) metres, on a particular day by the equation $$H = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) hours is the number of hours after midday.
(c) Calculate the maximum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this maximum occurs.
(d) Calculate, to the nearest minute, the times when the height of sea water is predicted, by this model, to be 7 metres.

1. (a) Express \(2 \cos 3 x - 3 \sin 3 x\) in the form \(R \cos ( 3 x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your answers to 3 significant figures.

$$\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } \cos 3 x$$ (b) Show that \(\mathrm { f } ^ { \prime } ( x )\) can be written in the form $$\mathrm { f } ^ { \prime } ( x ) = R \mathrm { e } ^ { 2 x } \cos ( 3 x + \alpha )$$ where \(R\) and \(\alpha\) are the constants found in part (a).
(c) Hence, or otherwise, find the smallest positive value of \(x\) for which the curve with equation \(y = \mathrm { f } ( x )\) has a turning point.

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

1. find the exact value of \(R\) and the value of \(\alpha\) to one decimal place.
2. Hence solve the equation $$7 \cos x + \sin x = 5$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
3. State the values of \(k\) for which the equation $$7 \cos x + \sin x = k$$ has only one solution in the interval \(0 \leqslant x < 360 ^ { \circ }\)

8. (a) Express \(9 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\) to 4 decimal places.
(b) (i) State the maximum value of \(9 \cos \theta - 2 \sin \theta\) (ii) Find the value of \(\theta\), for \(0 < \theta < 2 \pi\), at which this maximum occurs. Ruth models the height \(H\) above the ground of a passenger on a Ferris wheel by the equation $$H = 10 - 9 \cos \left( \frac { \pi t } { 5 } \right) + 2 \sin \left( \frac { \pi t } { 5 } \right)$$ where \(H\) is measured in metres and \(t\) is the time in minutes after the wheel starts turning. \includegraphics[max width=\textwidth, alt={}, center]{0f6fd881-4d4b-4f80-96cc-6da41cc33c60-14_572_458_719_1158}
(c) Calculate the maximum value of \(H\) predicted by this model, and the value of \(t\), when this maximum first occurs. Give your answers to 2 decimal places.
(d) Determine the time for the Ferris wheel to complete two revolutions.

8. \begin{figure}[h] \end{figure} Kate crosses a road, of constant width 7 m , in order to take a photograph of a marathon runner, John, approaching at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Kate is 24 m ahead of John when she starts to cross the road from the fixed point \(A\). John passes her as she reaches the other side of the road at a variable point \(B\), as shown in Figure 2.
Kate's speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she moves in a straight line, which makes an angle \(\theta\), \(0 < \theta < 150 ^ { \circ }\), with the edge of the road, as shown in Figure 2. You may assume that \(V\) is given by the formula $$V = \frac { 21 } { 24 \sin \theta + 7 \cos \theta } , \quad 0 < \theta < 150 ^ { \circ }$$

1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants and where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places. Given that \(\theta\) varies,
2. find the minimum value of \(V\). Given that Kate's speed has the value found in part (b),
3. find the distance \(A B\). Given instead that Kate's speed is \(1.68 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
4. find the two possible values of the angle \(\theta\), given that \(0 < \theta < 150 ^ { \circ }\).

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\), with equation \(y = 6 \cos x + 2.5 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\)

1. Express \(6 \cos x + 2.5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give your value of \(\alpha\) to 3 decimal places.
2. Find the coordinates of the points on the graph where the curve \(C\) crosses the coordinate axes. A student records the number of hours of daylight each Sunday throughout the year. She starts on the last Sunday in May with a recording of 18 hours, and continues until her final recording 52 weeks later. She models her results with the continuous function given by $$H = 12 + 6 \cos \left( \frac { 2 \pi t } { 52 } \right) + 2.5 \sin \left( \frac { 2 \pi t } { 52 } \right) , \quad 0 \leqslant t \leqslant 52$$ where \(H\) is the number of hours of daylight and \(t\) is the number of weeks since her first recording. Use this function to find
3. the maximum and minimum values of \(H\) predicted by the model,
4. the values for \(t\) when \(H = 16\), giving your answers to the nearest whole number.
   [0pt] [You must show your working. Answers based entirely on graphical or numerical methods are not acceptable.] \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-14_40_58_2460_1893}

9. (a) Express \(2 \sin \theta - 4 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the value of \(\alpha\) to 3 decimal places. $$H ( \theta ) = 4 + 5 ( 2 \sin 3 \theta - 4 \cos 3 \theta ) ^ { 2 }$$ Find
(b) (i) the maximum value of \(\mathrm { H } ( \theta )\),
(ii) the smallest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this maximum value occurs. Find
(c) (i) the minimum value of \(\mathrm { H } ( \theta )\),
(ii) the largest value of \(\theta\), for \(0 \leqslant \theta < \pi\), at which this minimum value occurs.

3. $$g ( \theta ) = 4 \cos 2 \theta + 2 \sin 2 \theta$$ Given that \(\mathrm { g } ( \theta ) = R \cos ( 2 \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\),

1. find the exact value of \(R\) and the value of \(\alpha\) to 2 decimal places.
2. Hence solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), $$4 \cos 2 \theta + 2 \sin 2 \theta = 1$$ giving your answers to one decimal place. Given that \(k\) is a constant and the equation \(\mathrm { g } ( \theta ) = k\) has no solutions,
3. state the range of possible values 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.

1. (a) Write \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 \leqslant \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and give the value of \(\alpha\) in radians to 3 decimal places.
(b) Show that the equation $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ can be rewritten in the form $$5 \cos 2 x - 2 \sin 2 x = c$$ where \(c\) is a positive constant to be determined.
(c) Hence or otherwise, solve, for \(0 \leqslant x < \pi\), $$5 \cot 2 x - 3 \operatorname { cosec } 2 x = 2$$ giving your answers to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

1. (a) Express \(\sin \theta - 2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

Give the exact value of \(R\) and the value of \(\alpha\), in radians, to 3 decimal places. $$\mathrm { M } ( \theta ) = 40 + ( 3 \sin \theta - 6 \cos \theta ) ^ { 2 }$$ (b) Find

1. the maximum value of \(\mathrm { M } ( \theta )\),
2. the smallest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { M } ( \theta )\) occurs. $$N ( \theta ) = \frac { 30 } { 5 + 2 ( \sin 2 \theta - 2 \cos 2 \theta ) ^ { 2 } }$$ (c) Find
3. the maximum value of \(\mathrm { N } ( \theta )\),
4. the largest value of \(\theta\), in the range \(0 < \theta \leqslant 2 \pi\), at which the maximum value of \(\mathrm { N } ( \theta )\) occurs.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

   END

1. In a particular circuit the current, \(I\) amperes, is given by

$$I = 4 \sin \theta - 3 \cos \theta , \quad \theta > 0$$ where \(\theta\) is an angle related to the voltage. Given that \(I = R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 \leqslant \alpha < 360 ^ { \circ }\),

1. find the value of \(R\), and the value of \(\alpha\) to 1 decimal place.
2. Hence solve the equation \(4 \sin \theta - 3 \cos \theta = 3\) to find the values of \(\theta\) between 0 and \(360 ^ { \circ }\).
3. Write down the greatest value for \(I\).
4. Find the value of \(\theta\) between 0 and \(360 ^ { \circ }\) at which the greatest value of \(I\) occurs.
   8. continued

7. (i) Given that \(y = \tan x + 2 \cos x\), find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { \pi } { 4 }\).
(ii) Given that \(x = \tan \frac { 1 } { 2 } y\), prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 1 + x ^ { 2 } }\).
(iii) Given that \(y = \mathrm { e } ^ { - x } \sin 2 x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(R \mathrm { e } ^ { - x } \cos ( 2 x + \alpha )\). Find, to 3 significant figures, the values of \(R\) and \(\alpha\), where \(0 < \alpha < \frac { \pi } { 2 }\).

$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$ 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 $$14 \cos ^ { 2 } x - 48 \sin x \cos x$$

7. (i) Express \(2 \sin x ^ { \circ } - 3 \cos x ^ { \circ }\) in the form \(R \sin ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(ii) Show that the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ can be written in the form $$2 \sin x ^ { \circ } - 3 \cos x ^ { \circ } = 1$$ (iii) 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.

1. (i) Express \(4 \sin x + 3 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
   (ii) State the minimum value of \(4 \sin x + 3 \cos x\) and the smallest positive value of \(x\) for which this minimum value occurs.
   (iii) 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.