1.05f Trigonometric function graphs: symmetries and periodicities

7

1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leqslant x \leqslant 2 \pi\). State the values of the intercepts with the coordinate axes.
2. 1. Given that $$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$ prove that \(\cos \theta = \frac { 1 } { 2 }\).
   2. Hence solve the equation $$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$ in the interval \(0 \leqslant x \leqslant \pi\), giving your answers in radians to three significant figures.

6

1. Sketch, on the axes given below, the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = \sin 5 x\).
3. Describe the single geometrical transformation that maps the graph of \(y = \sin 5 x\) onto the graph of \(y = \sin \left( 5 x + 10 ^ { \circ } \right)\).
   [0pt] [2 marks]
   1. \includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-12_675_1417_906_370}

4. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180$$

1. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\) axis.
2. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
3. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\) [0pt] [P1 June 2002 Question 5]

6. $$f ( x ) = \cos 2 x , \quad 0 \leq x \leq \pi .$$

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Write down the coordinates of any points where the curve \(y = \mathrm { f } ( x )\) meets the coordinate axes.
3. Solve the equation \(\mathrm { f } ( x ) = 0.5\), giving your answers in terms of \(\pi\).

3. The vertical motion of a point on the surface of the water in a certain harbour may be modelled as Simple Harmonic Motion about a mean level. The diagram shows that, on a particular day, the depth of water in the harbour at low tide is 2 m and the depth of the water in the harbour at high tide is 10 m . The table below shows the times of high and low tides on this day. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-4_405_912_621_233}

1. Write down the period and amplitude of the motion.
2. Let \(x \mathrm {~m}\) denote the height of water above mean level \(t\) hours after 5a.m. Find an expression for \(x\) in terms of \(t\).
3. The depth of water must be at least 4 m for boats to safely use the harbour. Determine the earliest time, after low tide at 5 a.m., at which boats can safely leave the harbour and hence find the latest possible time of return before the next low tide.
4. Calculate the rate at which the level of water is falling at 2 p.m.

3 In this question you must show detailed reasoning. You are given that \(\tan 30 ^ { \circ } = \frac { 1 } { \sqrt { 3 } }\).
Explain why \(\tan 690 ^ { \circ } = - \frac { 1 } { \sqrt { 3 } }\).

11 The height, in metres, of the sea at a coastal town during a day may be modelled by the function $$\mathrm { f } ( t ) = 1.7 + 0.8 \sin ( 30 t ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.

1. (a) Find the maximum height of the sea as given by this model.
   (b) Find the time of day at which this maximum height first occurs.
2. Determine the time when, according to this model, the height of the sea will first be 1.2 m . The height, in metres, at a different coastal town during a day may be modelled by the function $$\mathrm { g } ( t ) = a + b \sin ( c t + d ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.
3. It is given that at this different coastal town the maximum height of the sea is 3.1 m , and this height occurs at 0500 and 1700. The minimum height of the sea is 0.7 m , and this height occurs at 1100 and 2300 . Find the values of the constants \(a , b , c\) and \(d\).
4. It is instead given that the maximum height of the sea actually occurs at 0500 and 1709 . State, with a reason, how this will affect the value of \(c\) found in part (iii). \includegraphics[max width=\textwidth, alt={}, center]{74a37bca-0b28-4c48-bd21-a9304f31b8f8-6_563_568_322_751} The diagram shows the curve \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\) for \(x \geqslant 0\).
5. Use the substitution \(u ^ { 2 } = x + 1\) to find \(\int \mathrm { e } ^ { \sqrt { x + 1 } } \mathrm {~d} x\).
6. Make \(x\) the subject of the equation \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\).
7. Hence show that \(\int _ { \mathrm { e } } ^ { \mathrm { e } ^ { 4 } } \left( ( \ln y ) ^ { 2 } - 1 \right) \mathrm { d } y = 9 \mathrm { e } ^ { 4 }\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA

8

1. Solve the equation \(\cos x = 0.3\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
2. The diagram shows the graph of \(y = \cos x\) for \(0 \leqslant x \leqslant 2 \pi\) and the line \(y = k\). \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-5_524_805_559_648} The line \(y = k\) intersects the curve \(y = \cos x , 0 \leqslant x \leqslant 2 \pi\), at the points \(P\) and \(Q\). The point \(M\) is the minimum point of the curve.
   1. Write down the coordinates of the point \(M\).
   2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of \(Q\) in terms of \(\pi\) and \(\alpha\).
3. Describe the geometrical transformation that maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
4. Solve the equation \(\cos 2 x = \cos \frac { 4 \pi } { 5 }\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving the values of \(x\) in terms of \(\pi\).
   (4 marks)

7

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Write down the two solutions of the equation \(\tan x = \tan 61 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. 1. Given that \(\sin \theta + \cos \theta = 0\), show that \(\tan \theta = - 1\).
   2. Hence solve the equation \(\sin \left( x - 20 ^ { \circ } \right) + \cos \left( x - 20 ^ { \circ } \right) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
4. Describe the single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x - 20 ^ { \circ } \right)\).
5. The curve \(y = \tan x\) is stretched in the \(x\)-direction with scale factor \(\frac { 1 } { 4 }\) to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

2 One of the diagrams below shows the graph of \(y = \sin \left( x + 90 ^ { \circ } \right)\) for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) Identify the correct graph. Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_451_465_568_497} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_124_154_724_1073} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_458_472_1105_495}
□ \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_453_468_1647_497} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_117_132_1809_1091} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_461_479_2183_488}

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 $$\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } } \equiv \frac { 1 } { x + 1 } - \frac { 6 } { ( x + 2 ) ^ { 2 } }$$

7 Sketch the graph of $$y = \cot \left( x - \frac { \pi } { 2 } \right)$$ for \(0 \leq x \leq 2 \pi\) [0pt] [3 marks] \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-08_1650_1226_587_408} \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-09_2488_1716_219_153}

7. \begin{figure}[h] \end{figure} Figure 3 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = - 4 \cos x$$ where \(x\) is measured in radians.
Points \(P\) and \(Q\) lie on the curve and are shown in Figure 3.

1. State
   1. the coordinates of \(P\)
   2. the coordinates of \(Q\) The curve \(C _ { 2 }\) has equation \(y = - 4 \cos x + k\) where \(x\) is measured in radians and \(k\) is a constant. Given that \(C _ { 2 }\) has a maximum \(y\) value of 11
2. 1. state the value of \(k\)
   2. state the coordinates of the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate. On the opposite page there is a copy of Figure 3 labelled Diagram 1.
3. Using Diagram 1, state the number of solutions of the equation $$- 4 \cos x = 5 - \frac { 10 } { \pi } x$$ giving a reason for your answer. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-23_860_1016_1676_529} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

6

1. Sketch the graph of \(y = \cos 2 x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).

4

1. Sketch the graph of \(y = \sqrt { 2 } \sin x\) for \(0 \leqslant x \leqslant 2 \pi\). The points \(P\) and \(Q\) on the graph have \(x\)-coordinates \(\frac { 1 } { 4 } \pi\) and \(\frac { 3 } { 4 } \pi\), respectively.
2. Determine the equation of the tangent to the curve at \(P\). The normals to the curve at \(P\) and \(Q\) intersect at the point \(R\).
3. Determine the exact coordinates of \(R\).

\includegraphics{figure_8} The diagram shows the graph of \(y = f(x)\) where the function \(f\) is defined by $$f(x) = 3 + 2\sin \frac{1}{4}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$

1. On the diagram above, sketch the graph of \(y = f^{-1}(x)\). [2]
2. Find an expression for \(f^{-1}(x)\). [2]
3. \includegraphics{figure_8c} The diagram above shows part of the graph of the function \(g(x) = 3 + 2\sin \frac{1}{4}x\) for \(-2\pi \leqslant x \leqslant 2\pi\). Complete the sketch of the graph of \(g(x)\) on the diagram above and hence explain whether the function \(g\) has an inverse. [2]
4. Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\) to the graph of \(y = f(x)\), making clear the order in which the transformations are applied. [6]

\includegraphics{figure_1} The diagram shows the curve with equation \(y = a\sin(bx) + c\) for \(0 \leqslant x \leqslant 2\pi\), where \(a\), \(b\) and \(c\) are positive constants.

1. State the values of \(a\), \(b\) and \(c\). [3]
2. For these values of \(a\), \(b\) and \(c\), determine the number of solutions in the interval \(0 \leqslant x \leqslant 2\pi\) for each of the following equations:
   1. \(a\sin(bx) + c = 7 - x\) [1]
   2. \(a\sin(bx) + c = 2\pi(x - 1)\). [1]

The function \(f : x \mapsto 4 - 3\sin x\) is defined for the domain \(0 \leq x < 2\pi\).

1. Solve the equation \(f(x) = 2\). [3]
2. Sketch the graph of \(y = f(x)\). [2]
3. Find the set of values of \(k\) for which the equation \(f(x) = k\) has no solution. [2]

The function \(g : x \mapsto 4 - 3\sin x\) is defined for the domain \(\frac{1}{2}\pi \leq x \leq A\).

1. State the largest value of \(A\) for which \(g\) has an inverse. [1]
2. For this value of \(A\), find the value of \(g^{-1}(3)\). [2]

The function \(f\) is such that \(f(x) = 3 - 4\cos^k x\), for \(0 \leq x \leq \pi\), where \(k\) is a constant.

1. In the case where \(k = 2\),
   1. find the range of \(f\), [2]
   2. find the exact solutions of the equation \(f(x) = 1\). [3]
2. In the case where \(k = 1\),
   1. sketch the graph of \(y = f(x)\), [2]
   2. state, with a reason, whether \(f\) has an inverse. [1]

The function \(\text{f} : x \mapsto 5 + 3\cos(\frac{1}{3}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).

1. Solve the equation \(\text{f}(x) = 7\), giving your answer correct to 2 decimal places. [3]
2. Sketch the graph of \(y = \text{f}(x)\). [2]
3. Explain why \(\text{f}\) has an inverse. [1]
4. Obtain an expression for \(\text{f}^{-1}(x)\). [3]

A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, \(h\) m, of a passenger above the ground is given by the formula \(h = 60(1 - \cos kt)\). In this formula, \(k\) is a constant, \(t\) is the time in minutes that has elapsed since the passenger started the ride at ground level and \(kt\) is measured in radians.

1. Find the greatest height of the passenger above the ground. [1]

One complete revolution of the wheel takes 30 minutes.

1. Show that \(k = \frac{\pi}{15}\pi\). [2]
2. Find the time for which the passenger is above a height of 90 m. [3]

\includegraphics{figure_9} The function f : \(x \mapsto p \sin^2 2x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \text{f}(x)\).

1. In terms of \(p\) and \(q\), state the range of f. [2]
2. State the number of solutions of the following equations.
   1. \(\text{f}(x) = p + q\) [1]
   2. \(\text{f}(x) = q\) [1]
   3. \(\text{f}(x) = \frac{1}{2}p + q\) [1]
3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\text{f}(x) = 4\), showing all necessary working. [5]

Functions f and g are defined by $$f : x \mapsto 2 - 3\cos x \text{ for } 0 \leqslant x \leqslant 2\pi,$$ $$g : x \mapsto \frac{1}{2}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$

1. Solve the equation \(\text{fg}(x) = 1\). [3]
2. Sketch the graph of \(y = \text{f}(x)\). [3]