1.05o Trigonometric equations: solve in given intervals

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

Give the exact value of \(R\) and the value of \(\alpha\) in radians to 3 decimal places. \begin{figure}[h] \end{figure} Figure 6 shows the cross-section of a water wheel.
The wheel is free to rotate about a fixed axis through the point \(C\).
The point \(P\) is at the end of one of the paddles of the wheel, as shown in Figure 6.
The water level is assumed to be horizontal and of constant height.
The vertical height, \(H\) metres, of \(P\) above the water level is modelled by the equation $$H = 3 + 4 \cos ( 0.5 t ) - 2 \sin ( 0.5 t )$$ where \(t\) is the time in seconds after the wheel starts rotating.
Using the model, find
(b) (i) the maximum height of \(P\) above the water level,
(ii) the value of \(t\) when this maximum height first occurs, giving your answer to one decimal place. In a single revolution of the wheel, \(P\) is below the water level for a total of \(T\) seconds. According to the model,
(c) find the value of \(T\) giving your answer to 3 significant figures.
(Solutions based entirely on calculator technology are not acceptable.) In reality, the water level may not be of constant height.
(d) Explain how the equation of the model should be refined to take this into account.

2. Some A level students were given the following question. Solve, for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), the equation $$\cos \theta = 2 \sin \theta$$ The attempts of two of the students are shown below. Student \(B\) $$\begin{aligned} \cos \theta & = 2 \sin \theta \\ \cos ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta \\ 1 - \sin ^ { 2 } \theta & = 4 \sin ^ { 2 } \theta \\ \sin ^ { 2 } \theta & = \frac { 1 } { 5 } \\ \sin \theta & = \pm \frac { 1 } { \sqrt { 5 } } \\ \theta & = \pm 26.6 ^ { \circ } \end{aligned}$$

1. Identify an error made by student \(A\). Student \(B\) gives \(\theta = - 26.6 ^ { \circ }\) as one of the answers to \(\cos \theta = 2 \sin \theta\).
2. 1. Explain why this answer is incorrect.
   2. Explain how this incorrect answer arose.

1. (a) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation

$$3 \sin ^ { 2 } x + \sin x + 8 = 9 \cos ^ { 2 } x$$ giving your answers to 2 decimal places.
(b) Hence find the smallest positive solution of the equation $$3 \sin ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) + \sin \left( 2 \theta - 30 ^ { \circ } \right) + 8 = 9 \cos ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right)$$ giving your answer to 2 decimal places.

13. (a) Show that $$\operatorname { cosec } 2 x + \cot 2 x \equiv \cot x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$ (b) Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), $$\operatorname { cosec } \left( 4 \theta + 10 ^ { \circ } \right) + \cot \left( 4 \theta + 10 ^ { \circ } \right) = \sqrt { 3 }$$ You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

3 In this question you must show detailed reasoning.

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 8 x + 3\).
   1. Show that \(f ( 1 ) = 0\).
   2. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Hence solve the equation \(2 \sin ^ { 3 } \theta + 3 \sin ^ { 2 } \theta - 8 \sin \theta + 3 = 0\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

3 In this question you must show detailed reasoning.

1. Solve the equation \(4 \sin ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
2. Prove that \(\frac { \sin ^ { 2 } \theta - 1 + \cos \theta } { 1 - \cos \theta } \equiv \cos \theta \quad ( \cos \theta \neq 1 )\).

4 In this question you must show detailed reasoning. Solve the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. \(2 \tan x + 1 = 4\)
2. \(5 \sin x - 1 = 2 \cos ^ { 2 } x\)

7

1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos ^ { 2 } x + 5 \cos x - 2 = 0$$
2. Hence solve the equation $$2 \sin 2 \theta \tan 2 \theta = \cos 2 \theta + 5$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), correct to 1 decimal place.

5

1. Show that the equation \(2 \cos x \tan ^ { 2 } x = 3 ( 1 + \cos x )\) can be expressed in the form $$5 \cos ^ { 2 } x + 3 \cos x - 2 = 0$$ \section*{(b) In this question you must show detailed reasoning.} Hence solve the equation $$2 \cos 3 \theta \tan ^ { 2 } 3 \theta = 3 ( 1 + \cos 3 \theta ) ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(120 ^ { \circ }\), correct to \(\mathbf { 1 }\) decimal place where appropriate.

3 A Ferris wheel at a fairground rotates in a vertical plane. The height above the ground of a seat on the wheel is \(h\) metres at time \(t\) seconds after the seat is at its lowest point. The height is given by the equation \(h = 15 - 14 \cos ( k t ) ^ { \circ }\), where \(k\) is a positive constant.

1. 1. Write down the greatest height of a seat above the ground.
   2. Write down the least height of a seat above the ground.
2. Given that a seat first returns to its lowest point after 150 seconds, calculate the value of \(k\).
3. Determine for how long a seat is 20 metres or more above the ground during one complete revolution. Give your answer correct to the nearest tenth of a second.

5

1. Sketch the graphs of \(y = 4 \cos x\) and \(y = 2 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) on the same axes.
2. Find the exact coordinates of the point of intersection of these graphs, giving your answer in the form (arctan \(a , k \sqrt { b }\) ), where \(a\) and \(b\) are integers and \(k\) is rational.
3. A student argues that without the condition \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) all the points of intersection of the graphs would occur at intervals of \(360 ^ { \circ }\) because both \(\sin x\) and \(\cos x\) are periodic functions with this period. Comment on the validity of the student's argument.

5 Part of the graph of \(y = f ( x )\) is shown below. The graph is the image of \(y = \tan x ^ { \circ }\) after a stretch in the \(x\)-direction. \includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-4_791_1022_1014_244}

1. Find the equation of the graph.
2. Write down the period of the function \(\mathrm { f } ( x )\).
3. In this question you must show detailed reasoning. Find all the roots of the equation \(\mathrm { f } ( x ) = 1\) for \(0 ^ { \circ } \leqslant x ^ { \circ } \leqslant 360 ^ { \circ }\).

4 In this question you must show detailed reasoning.
Solve the equation \(6 \cos ^ { 2 } x + \sin x = 5\), giving all the roots in the interval \(- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

14 In this question you must show detailed reasoning.
Solve the equation \(5 - \cos \theta - 6 \sin ^ { 2 } \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\). Turn over for question 15

8 In this question you must show detailed reasoning.
Solve the equation \(3 \cos \theta + 8 \tan \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your answers correct to the nearest degree.

6

1. The graph of \(y = 3 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) is shown in Fig. 6.
   On the copy of Fig. 6 in the Printed Answer Booklet, sketch the graph of \(y = 2 \cos \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-05_818_1507_571_351} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. In this question you must show detailed reasoning. Determine the values of \(\theta , 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\), for which the two graphs cross.

3 In this question you must show detailed reasoning.
Solve the equation \(\sec ^ { 2 } \theta + 2 \tan \theta = 4\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

6

1. Prove that \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = \cot \theta\).
2. Hence find the exact roots of the equation \(\frac { \sin \theta } { 1 - \cos \theta } - \frac { 1 } { \sin \theta } = 3 \tan \theta\) in the interval \(0 \leqslant \theta \leqslant \pi\). Answer all the questions.
   Section B (75 marks)

3

1. Sketch the graph of \(\mathrm { y } = \arctan \mathrm { x }\) where \(x\) is in radians.
2. In this question you must show detailed reasoning. Find all points of intersection of the curves \(\mathrm { y } = 3 \sin \mathrm { xcos } \mathrm { x }\) and \(\mathrm { y } = \cos ^ { 2 } \mathrm { x }\) for \(- \pi \leqslant x \leqslant \pi\).

6

1. Show that the equation \(\sin \left( x + \frac { 1 } { 6 } \pi \right) = \cos \left( x - \frac { 1 } { 4 } \pi \right)\) can be written in the form \(\tan x = \frac { \sqrt { 2 } - 1 } { \sqrt { 3 } - \sqrt { 2 } }\).
2. Hence solve the equation \(\sin \left( x + \frac { 1 } { 6 } \pi \right) = \cos \left( x - \frac { 1 } { 4 } \pi \right)\) for \(0 \leqslant x \leqslant 2 \pi\).

10 The diagram shows the graph of \(\mathrm { y } = 1.5 + \sin ^ { 2 } \mathrm { x }\) for \(0 \leqslant x \leqslant 2 \pi\). \includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-07_512_1278_322_242}

1. Show that the equation of the graph can be written in the form \(\mathrm { y } = \mathrm { a } - \mathrm { b } \cos 2 \mathrm { x }\) where \(a\) and \(b\) are constants to be determined.
2. Write down the period of the function \(1.5 + \sin ^ { 2 } x\).
3. Determine the \(x\)-coordinates of the points of intersection of the graph of \(y = 1.5 + \sin ^ { 2 } x\) with the graph of \(\mathrm { y } = 1 + \cos 2 \mathrm { x }\) in the interval \(0 \leqslant x \leqslant 2 \pi\).

4

1. On the axes in the Printed Answer Booklet, sketch the graph of \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve the equation \(\sin 2 \theta = - \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\). \(5 M\) is the event that an A-level student selected at random studies mathematics. \(C\) is the event that an A-level student selected at random studies chemistry.
   You are given that \(\mathrm { P } ( M ) = 0.42 , \mathrm { P } ( C ) = 0.36\) and \(\mathrm { P } ( \mathrm { M }\) and \(\mathrm { C } ) = 0.24\). These probabilities are shown in the two-way table below.

   \cline { 2 - 4 } \multicolumn{1}{c|}{}	\(M\)	\(M ^ { \prime }\)	Total
   \(C\)	0.24		0.36
   \(C ^ { \prime }\)
   Total	0.42		1

4 In this question you must show detailed reasoning.
Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).

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.

2 Solve the equation \(\sin 2 x = 0.3\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\). Give your answer(s) correct to \(\mathbf { 1 }\) decimal place.