1.05f Trigonometric function graphs: symmetries and periodicities

5

1. Sketch, on the same diagram, the graphs of \(y = \sin x\) and \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
2. Verify that \(x = 30 ^ { \circ }\) is a root of the equation \(\sin x = \cos 2 x\), and state the other root of this equation for which \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
3. Hence state the set of values of \(x\), for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\), for which \(\sin x < \cos 2 x\).

10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).

1. Find the range of f .
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
4. State the largest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

7

1. \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-12_499_568_267_826} The diagram shows part of the graph of \(y = a + b \sin x\). Find the values of the constants \(a\) and \(b\).
2. 1. Show that the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ may be expressed as \(3 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0\).
   2. Hence solve the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

6

1. The function f , defined by \(\mathrm { f } : x \mapsto a + b \sin x\) for \(x \in \mathbb { R }\), is such that \(\mathrm { f } \left( \frac { 1 } { 6 } \pi \right) = 4\) and \(\mathrm { f } \left( \frac { 1 } { 2 } \pi \right) = 3\).
   1. Find the values of the constants \(a\) and \(b\).
   2. Evaluate \(\mathrm { ff } ( 0 )\).
2. The function g is defined by \(\mathrm { g } : x \mapsto c + d \sin x\) for \(x \in \mathbb { R }\). The range of g is given by \(- 4 \leqslant \mathrm {~g} ( x ) \leqslant 10\). Find the values of the constants \(c\) and \(d\).

5. \begin{figure}[h] \end{figure} Figure 2 shows a plot of part of the curve with equation \(y = \cos 2 x\) with \(x\) being measured in radians. The point \(P\), shown on Figure 2, is a minimum point on the curve.

1. State the coordinates of \(P\). A copy of Figure 2, called Diagram 1, is shown at the top of the next page.
2. Sketch, on Diagram 1, the curve with equation \(y = \sin x\)
3. Hence, or otherwise, deduce the number of solutions of the equation
   1. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 20 \pi\)
   2. \(\cos 2 x = \sin x\) that lie in the region \(0 \leqslant x \leqslant 21 \pi\) \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f8d35d-c2dd-4a1f-a4bb-a4fa06413d12-11_693_1050_301_447} \captionsetup{labelformat=empty} \caption{
      Diagram 1}\}
      \end{figure} \textbackslash section*\{Diagram 1

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C _ { 1 }\) with equation \(y = 3 \sin x\), where \(x\) is measured in degrees. The point \(P\) and the point \(Q\) lie on \(C _ { 1 }\) and are shown in Figure 3.

1. State
   1. the coordinates of \(P\),
   2. the coordinates of \(Q\). A different curve \(C _ { 2 }\) has equation \(y = 3 \sin x + k\), where \(k\) is a constant.
      The curve \(C _ { 2 }\) has a maximum \(y\) value of 10
      The point \(R\) is the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate.
2. State the coordinates of \(R\). Figure 3

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C _ { 1 }\) with equation \(y = 4 \cos x ^ { \circ }\) The point \(P\) and the point \(Q\) lie on \(C _ { 1 }\) and are shown in Figure 1.

1. State
   1. the coordinates of \(P\),
   2. the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 4 \cos x ^ { \circ } + k\), where \(k\) is a constant.
      Curve \(C _ { 2 }\) has a minimum \(y\) value of - 1
      The point \(R\) is the maximum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate.
2. State the coordinates of \(R\).

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the curve with equation $$y = A \cos ( x - 30 ) ^ { \circ }$$ where \(A\) is a constant. The point \(P\) is a minimum point on the curve and has coordinates \(( 30 , - 3 )\) as shown in Figure 4.

1. Write down the value of \(A\). The point \(Q\) is shown in Figure 4 and is a maximum point.
2. Find the coordinates of \(Q\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of

• the curve with equation \(y = \tan x\)
• the straight line l with equation \(y = \pi x\) in the interval \(- \pi < x < \pi\)

6. \begin{figure}[h] \end{figure} Figure 2 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = 5 \cos x$$ with \(x\) being measured in degrees.
The point \(P\), shown in Figure 2, is a minimum point on \(C _ { 1 }\)

1. State the coordinates of \(P\) The point \(Q\) lies on a different curve \(C _ { 2 }\) Given that point \(Q\)
   • is a maximum point on the curve
   • is the maximum point with the smallest \(x\) coordinate, \(x > 0\)
   • find the coordinates of \(Q\) when
     1. \(C _ { 2 }\) has equation \(y = 5 \cos x - 2\)
     2. \(C _ { 2 }\) has equation \(y = - 5 \cos x\)

9. \begin{figure}[h] \end{figure} Figure 3 shows a plot of the curve with equation \(y = \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)

1. State the coordinates of the minimum point on the curve with equation $$y = 4 \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }$$ A copy of Figure 3, called Diagram 1, is shown on the next page.
2. On Diagram 1, sketch and label the curves
   1. \(y = 1 + \sin \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
   2. \(y = \tan \theta , \quad 0 \leqslant \theta \leqslant 360 ^ { \circ }\)
3. Hence find the number of solutions of the equation
   1. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 2160 ^ { \circ }\)
   2. \(\tan \theta = 1 + \sin \theta\) that lie in the region \(0 \leqslant \theta \leqslant 1980 ^ { \circ }\)
      \includegraphics[max width=\textwidth, alt={}]{5eee32af-9b0e-428c-bbc6-1feef44e0e1e-25_746_808_577_575}
      \section*{Diagram 1}

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$y = \tan x \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ The line \(l\), shown in Figure 4, is an asymptote to \(y = \tan x\)

1. State an equation for \(l\). A copy of Figure 4, labelled Diagram 1, is shown on the next page.
2. 1. On Diagram 1, sketch the curve with equation $$y = \frac { 1 } { x } + 1 \quad - 2 \pi \leqslant x \leqslant 2 \pi$$ stating the equation of the horizontal asymptote of this curve.
   2. Hence, giving a reason, state the number of solutions of the equation
3. State the number of solutions of the equation \(\tan x = \frac { 1 } { x } + 1\) in the region
   1. \(0 \leqslant x \leqslant 40 \pi\)
   2. \(- 10 \pi \leqslant x \leqslant \frac { 5 } { 2 } \pi\) $$\begin{aligned} & \qquad \tan x = \frac { 1 } { x } + 1 \\ & \text { in the region } - 2 \pi \leqslant x \leqslant 2 \pi \end{aligned}$$" \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-31_725_1047_1078_447} \captionsetup{labelformat=empty} \caption{Diagram 1}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-32_2644_1837_118_114}

9. \begin{figure}[h] \end{figure} Figure 4 shows part of the graph of the curve with equation \(y = \sin x\) Given that \(\sin \alpha = p\), where \(0 < \alpha < 90 ^ { \circ }\)

1. state, in terms of \(p\), the value of
   1. \(2 \sin \left( 180 ^ { \circ } - \alpha \right)\)
   2. \(\sin \left( \alpha - 180 ^ { \circ } \right)\)
   3. \(3 + \sin \left( 180 ^ { \circ } + \alpha \right)\) A copy of Figure 4, labelled Diagram 1, is shown on page 27. On Diagram 1,
2. sketch the graph of \(y = \sin 2 x\)
3. Hence find, in terms of \(\alpha\), the \(x\) coordinates of any points in the interval \(0 < x < 180 ^ { \circ }\) where $$\sin 2 x = p$$
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-27_433_1331_296_310}
   \section*{Diagram 1}

\begin{figure}[h] \end{figure} Figure 3 shows part of the graph of the trigonometric function with equation \(y = \mathrm { f } ( x )\)

1. Write down an expression for \(\mathrm { f } ( x )\) On a separate diagram,
2. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { f } \left( x + \frac { \pi } { 4 } \right)\) Show clearly the coordinates of all the points where the curve intersects the coordinate axes.
   (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_378_1251_1617_408} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows part of the graph of the trigonometric function with equation \(y = \mathrm { g } ( x )\)
3. Write down an expression for \(\mathrm { g } ( x )\) On a separate diagram,
4. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { g } ( x ) - 2\) Show clearly the coordinates of the \(y\) intercept.

11. Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 12 \sin x$$ where \(x\) is measured in radians.
The point \(P\) shown in Figure 4 is a maximum point on \(C _ { 1 }\)

1. Find the coordinates of \(P\). The curve \(C _ { 2 }\) has equation $$y = 12 \sin x + k$$ where \(k\) is a constant.
   Given that the maximum value of \(y\) on \(C _ { 2 }\) is 3
2. find the coordinates of the minimum point on \(C _ { 2 }\) which has the smallest positive \(x\) coordinate. The curve \(C _ { 3 }\) has equation $$y = 12 \sin ( x + B )$$ where \(B\) is a positive constant.
   Given that \(\left( \frac { \pi } { 4 } , A \right)\), where \(A\) is a constant, is the minimum point on \(C _ { 3 }\) which has the smallest positive \(x\) coordinate,
3. find
   1. the value of \(A\),
   2. the smallest possible value of \(B\).

9. Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \sin \left( \frac { x } { 12 } \right)\), where \(x\) is measured in radians. The point \(M\) shown in Figure 5 is a minimum point on \(C\).

1. State the period of \(C\).
2. State the coordinates of \(M\). The smallest positive solution of the equation \(\sin \left( \frac { x } { 12 } \right) = k\), where \(k\) is a constant, is \(\alpha\). Find, in terms of \(\alpha\),
3. 1. the negative solution of the equation \(\sin \left( \frac { x } { 12 } \right) = k\) that is closest to zero,
   2. the smallest positive solution of the equation \(\cos \left( \frac { x } { 12 } \right) = k\).

5. (i) \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the points \(( - 5,0 )\) and \(( 0 , - 3 )\) and touches the \(x\)-axis at the point \(( 2,0 )\). On separate diagrams sketch the curve with equation

1. \(y = \mathrm { f } ( x + 2 )\)
2. \(y = \mathrm { f } ( - x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.
   (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_415_814_1548_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve with equation $$y = k \cos \left( x + \frac { \pi } { 6 } \right) \quad 0 \leqslant x \leqslant 2 \pi$$ where \(k\) is a constant.
   The curve meets the \(y\)-axis at the point \(( 0 , \sqrt { 3 } )\) and passes through the points \(( p , 0 )\) and ( \(q , 0\) ). Find
   1. the value of \(k\),
   2. the exact value of \(p\) and the exact value of \(q\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C _ { 1 }\) with equation $$y = 3 \cos \left( \frac { x } { n } \right) ^ { \circ } \quad x \geqslant 0$$ where \(n\) is a constant.
The curve \(C _ { 1 }\) cuts the positive \(x\)-axis for the first time at point \(P ( 270,0 )\), as shown in Figure 4.

1. 1. State the value of \(n\)
   2. State the period of \(C _ { 1 }\) The point \(Q\), shown in Figure 4, is a minimum point of \(C _ { 1 }\)
2. State the coordinates of \(Q\). The curve \(C _ { 2 }\) has equation \(y = 2 \sin x ^ { \circ } + k\), where \(k\) is a constant.
   The point \(R \left( a , \frac { 12 } { 5 } \right)\) and the point \(S \left( - a , - \frac { 3 } { 5 } \right)\), both lie on \(C _ { 2 }\) Given that \(a\) is a constant less than 90
3. find the value of \(k\).

10. The curve \(C\) has equation \(y = \cos \left( x - \frac { \pi } { 3 } \right) , 0 \leqslant x \leqslant 2 \pi\)

1. In the space below, sketch the curve \(C\).
2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
3. Solve, for \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\), $$\cos \left( x - \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number.

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \sin \left( x - 60 ^ { \circ } \right) , - 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\)

1. Write down the exact coordinates of the points at which \(C\) meets the two coordinate axes.
2. Solve, for \(- 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), $$4 \sin \left( x - 60 ^ { \circ } \right) = \sqrt { 6 } - \sqrt { 2 }$$ showing each stage of your working.

5. (a) Sketch the graph of \(y = \sin 2 x , \quad 0 \leqslant x \leqslant \frac { 3 \pi } { 2 }\) Show the coordinates of the points where your graph crosses the \(x\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = \sin 2 x\).
The values of \(y\) are rounded to 3 decimal places where necessary. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for

6. (a) Sketch the graph of \(y = 1 + \cos x , \quad 0 \leqslant x \leqslant 2 \pi\) Show on your sketch the coordinates of the points where your graph meets the coordinate axes.
(b) Use the trapezium rule, with 6 strips of equal width, to find an approximate value for $$\int _ { 0 } ^ { 2 \pi } ( 1 + \cos x ) d x$$

13. The height of sea water, \(h\) metres, on a harbour wall at time \(t\) hours after midnight is given by $$h = 3.7 + 2.5 \cos ( 30 t - 40 ) ^ { \circ } , \quad 0 \leqslant t < 24$$

1. Calculate the maximum value of \(h\) and the exact time of day when this maximum first occurs. Fishing boats cannot enter the harbour if \(h\) is less than 3
2. Find the times during the morning between which fishing boats cannot enter the harbour.
   Give these times to the nearest minute.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

10. The curve \(C\) has equation \(y = \sin \left( x + \frac { \pi } { 4 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\)

1. On the axes below, sketch the curve \(C\).
2. Write down the exact coordinates of all the points at which the curve \(C\) meets or intersects the \(x\)-axis and the \(y\)-axis.
3. Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-14_677_1031_1446_445}

15. The height of water, \(H\) metres, in a harbour on a particular day is given by the equation $$H = 4 + 1.5 \sin \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight, and \(\frac { \pi t } { 6 }\) is measured in radians.

1. Show that the height of the water at 1 a.m. is 4.75 metres.
2. Find the height of the water at 2 p.m.
3. Find, to the nearest minute, the first two times when the height of the water is 3 metres.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)