Trig Graphs & Exact Values

4

1. Sketch the curve \(y = 2 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. By adding a suitable straight line to your sketch, determine the number of real roots of the equation $$2 \pi \sin x = \pi - x$$ State the equation of the straight line.

6 The equation of a curve is \(y = 3 \cos 2 x\) and the equation of a line is \(2 y + \frac { 3 x } { \pi } = 5\).

1. State the smallest and largest values of \(y\) for both the curve and the line for \(0 \leqslant x \leqslant 2 \pi\).
2. Sketch, on the same diagram, the graphs of \(y = 3 \cos 2 x\) and \(2 y + \frac { 3 x } { \pi } = 5\) for \(0 \leqslant x \leqslant 2 \pi\).
3. State the number of solutions of the equation \(6 \cos 2 x = 5 - \frac { 3 x } { \pi }\) for \(0 \leqslant x \leqslant 2 \pi\).

1. (i) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
   (ii) Hence state how many solutions exist to the equation

$$\sin 2 x = \tan \frac { x } { 2 } ,$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.

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 \(Q\).

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 \(R\).

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

1 Sketch the graph of \(y = \sec x\), for \(0 \leqslant x \leqslant 2 \pi\).

2 The equation of a curve is \(y = 3 \cos 2 x\). The equation of a line is \(x + 2 y = \pi\). On the same diagram, sketch the curve and the line for \(0 \leqslant x \leqslant \pi\).

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}

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 \(- \pi \leqslant x \leqslant \pi\).

8. The length of the daylight, \(D ( t )\) in a town in Sweden can be modelled using the equation $$D ( t ) = 12 + 9 \sin \left( \frac { 360 t } { 365 } - 63.435 \right) \quad 0 \leq t \leq 365$$ where \(t\) is the number of days into the year, and the argument of \(\sin x\) is in degrees
a. Find the number of daylight hours after 90 days in that year.
b. Find the values of \(t\) when \(D ( t ) = 17\), giving your answers to the nearest integer. (Solutions based entirely on graphical or numerical methods are not acceptable)

3 Given that \(\cos \theta = \frac { 1 } { 3 }\) and \(\theta\) is acute, find the exact value of \(\tan \theta\).

1
\includegraphics[max width=\textwidth, alt={}, center]{62f7f1e2-a8e7-4574-a432-8e9b20b54d7a-2_750_1287_258_427} The diagram shows part of the graph of \(y = a + b \sin x\). State the values of the constants \(a\) and \(b\). [2

3 The function \(\mathrm { f } : x \mapsto a + b \cos x\) is defined for \(0 \leqslant x \leqslant 2 \pi\). Given that \(\mathrm { f } ( 0 ) = 10\) and that \(\mathrm { f } \left( \frac { 2 } { 3 } \pi \right) = 1\), find

1. the values of \(a\) and \(b\),
2. the range of \(f\),
3. the exact value of \(\mathrm { f } \left( \frac { 5 } { 6 } \pi \right)\).

3

1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).
   \(4 \theta\) is an acute angle and \(\sin \theta = \frac { 1 } { 4 }\). Find the exact value of \(\tan \theta\).

3 Find the general solution, in degrees, of the equation $$\cos \left( 5 x - 20 ^ { \circ } \right) = \cos 40 ^ { \circ }$$

1. Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which
   1. \(\cos ( \theta + 75 ) ^ { \circ } = 0\).
   2. \(\sin 2 \theta ^ { \circ } = 0.7\), giving your answers to one decima1 place.
      [0pt] [P1 January 2001 Question 3]
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9f1194cd-cc8a-4f8d-8010-c62fea344c4e-01_645_1408_1096_262} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Figure 1 shows the curve with equation \(y = 5 + 2 x - x ^ { 2 }\) and the line with equation \(y = 2\). The curve and the line intersect at the points \(A\) and \(B\).
2. Find the \(x\)-coordinates of \(A\) and \(B\).
   (3) The shaded region \(R\) is bounded by the curve and the line.
3. Find the area of \(R\).
   (6)

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

2 The graph of the function \(y = \cos \frac { 1 } { 2 } x\) for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) is one of the graphs shown below. Identify the correct graph.
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_373_634_671_502}
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\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_387_634_1133_502}
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_113_111_1265_1306}
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_366_629_1610_502}
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\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_368_629_2074_502}
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7. (a) Show that $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ may be expressed in the form $$12 \cos ^ { 2 } x + \cos x - 1 = 0$$ (b) Hence, using trigonometry, find all the solutions in the interval \(0 \leqslant x \leqslant 360 ^ { \circ }\) of $$12 \sin ^ { 2 } x - \cos x - 11 = 0$$ Give each solution, in degrees, to 1 decimal place.
\includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-15_106_97_2615_1784}

3 In this question you must show detailed reasoning. Given that \(5 \sin 2 x = 3 \cos x\), where \(0 ^ { \circ } < x < 90 ^ { \circ }\), find the exact value of \(\sin x\).

4. Find all values of \(x\) in the interval \(0 \leq x < 360 ^ { \circ }\) for which $$2 \sin ^ { 2 } x - 2 \cos x - \cos ^ { 2 } x = 1$$ giving non-exact answers to 1 decimal place.

1

1. State the exact value of \(\tan 300 ^ { \circ }\).
2. Express \(300 ^ { \circ }\) in radians, giving your answer in the form \(k \pi\), where \(k\) is a fraction in its lowest terms.

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

5

1. Sketch the graph of $$y = \sin 2 x$$ for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\)
   \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-05_1095_1246_534_402} 5
2. The equation $$\sin 2 x = A$$ has exactly two solutions for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\)
   State the possible values of \(A\).