Trig Graphs & Exact Values

$$f(x) = 5\sin 3x°, \quad 0 \leq x \leq 180.$$

1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis [3]
2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]

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

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)

12 One of the rides at a theme park is a room where the floor and ceiling both move up and down for \(10 \pi\) seconds. At time \(t\) seconds after the ride begins, the distance \(f\) metres of the floor above the ground is $$f = 1 - \cos t$$ At time \(t\) seconds after the ride begins, the distance \(c\) metres of the ceiling above the ground is $$c = 8 - 4 \sin t$$ The ride is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-16_448_766_932_635} 12

1. Show that the initial distance between the floor and ceiling is 8 metres.
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}]{6a03a035-ff32-4734-864b-a076aa9cbec0-17_2500_1721_214_148}

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.)

1. Sketch the graph of $$y = \sin 2x$$ for \(0° \leq x \leq 360°\) \includegraphics{figure_5a} [2 marks]
2. The equation $$\sin 2x = A$$ has exactly two solutions for \(0° \leq x \leq 360°\) State the possible values of \(A\). [1 mark]

Sketch, on separate diagrams, the graphs of the following functions for \(0 \leqslant x \leqslant 2\pi\) giving the coordinates of all points of intersection with the axes.

1. \(y = \sin x\). [1]
2. \(y = \sin\left(x + \frac{1}{6}\pi\right)\). [2]

5 The equation of a curve is \(y = 2 \cos x\).

1. Sketch the graph of \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\), stating the coordinates of the point of intersection with the \(y\)-axis. Points \(P\) and \(Q\) lie on the curve and have \(x\)-coordinates of \(\frac { 1 } { 3 } \pi\) and \(\pi\) respectively.
2. Find the length of \(P Q\) correct to 1 decimal place.
   The line through \(P\) and \(Q\) meets the \(x\)-axis at \(H ( h , 0 )\) and the \(y\)-axis at \(K ( 0 , k )\).
3. Show that \(h = \frac { 5 } { 9 } \pi\) and find the value of \(k\).

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

Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for

1. \(\cos \theta\), [1]
2. \(\tan \theta\), [2]
3. \(\sin(\theta + \pi)\). [1]

5. A cotangent function \(\cot x\) is defined as \(\cot x = \frac { \cos x } { \sin x } , x \neq 180 ^ { \circ } k , k \in \mathbb { Z }\).
a) If \(- 270 ^ { \circ } \leq \alpha \leq - 180 ^ { \circ }\) and \(\cot \alpha = - \frac { 12 } { 5 }\), find the exact value of \(\sin \alpha\) and \(\cos \alpha\).
b) If the sum of the squares of the side lengths of a triangle equals 2021 and the sum of the cotangents of its angles is 43 , find the area of that triangle.
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]
[0pt] [Question 5 - Continued]

5 \includegraphics[max width=\textwidth, alt={}, center]{f759ce41-708e-4fe7-80b9-adc2be2972ac-08_526_499_258_824} The diagram shows the graphs of \(y = \tan x\) and \(y = \cos x\) for \(0 \leqslant x \leqslant \pi\). The graphs intersect at points \(A\) and \(B\).

1. Find by calculation the \(x\)-coordinate of \(A\).
2. Find by calculation the coordinates of \(B\).

9 \includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-4_406_625_248_721} The diagram shows part of the curve \(y = 2 \cos \frac { 1 } { 3 } x\), where \(x\) is in radians, and the line \(y = k\).

1. The smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\),
   1. the next smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\),
   2. the smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = - k\).
   3. The curve \(y = 2 \cos \frac { 1 } { 3 } x\) is shown in the Printed Answer Book. On the diagram, and for the same values of \(x\), sketch the curve of \(y = \sin \frac { 1 } { 3 } x\).
   4. Calculate the \(x\)-coordinates of the points of intersection of the curves in part (ii). Give your answers in radians correct to 3 significant figures. \section*{END OF QUESTION PAPER}

7. \(\left[ \arccos x \right.\) and \(\arctan x\) are alternative notation for \(\cos ^ { - 1 } x\) and \(\tan ^ { - 1 } x\) respectively \(]\) \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \cos ( \cos x ) , 0 \leqslant x < 2 \pi\) .
The curve has turning points at \(( 0 , \cos 1 ) , P , Q\) and \(R\) as shown in Figure 2.

1. Find the coordinates of the points \(P , Q\) and \(R\) . The curve \(C _ { 2 }\) has equation \(y = \sin ( \cos x ) , 0 \leqslant x < 2 \pi\) .The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(S\) and \(T\) .
2. Copy Figure 2 and on this diagram sketch \(C _ { 2 }\) stating the coordinates of the minimum point on \(C _ { 2 }\) and the points where \(C _ { 2 }\) meets or crosses the coordinate axes. The coordinates of \(S\) are \(( \alpha , d )\) where \(0 < \alpha < \pi\) .
3. Show that \(\alpha = \arccos \left( \frac { \pi } { 4 } \right)\) .
4. Find the value of \(d\) in surd form and write down the coordinates of \(T\) . The tangent to \(C _ { 1 }\) at the point \(S\) has gradient \(\tan \beta\) .
5. Show that \(\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)\) .
6. Find,in terms of \(\beta\) ,the obtuse angle between the tangent to \(C _ { 1 }\) at \(S\) and the tangent to \(C _ { 2 }\) at \(S\) .

4

1. Sketch and label, on the same diagram, the graphs of \(y = 2 \sin x\) and \(y = \cos 2 x\), for the interval \(0 \leqslant x \leqslant \pi\).
2. Hence state the number of solutions of the equation \(2 \sin x = \cos 2 x\) in the interval \(0 \leqslant x \leqslant \pi\).

1. Sketch the graph of \(y = \cos x\) for \(0° \leq x \leq 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leq x \leq 360°\). Label each graph clearly. [3]
2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leq x \leq 360°\). [2]

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.

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

1. Fig. 5 shows the graph of a sine function. \includegraphics{figure_5} State the equation of this curve. [2]
2. Sketch the graph of \(y = \sin x - 3\) for \(0° \leq x \leq 450°\). [2]

4 \includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-05_677_1591_260_278} The diagram shows part of the graph of \(y = a \tan ( x - b ) + c\).
Given that \(0 < b < \pi\), state the values of the constants \(a , b\) and \(c\).

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 \(0 \leqslant x \leqslant \pi\).

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 \(\mathrm { f } \left( \frac { 5 } { 6 } \pi \right)\).

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 \(y = \cos 2 x\).

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

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}

State the interval for which \(\sin x\) is a decreasing function for \(0° \leq x \leq 360°\) [2 marks]

Sketch the curve \(y = 2\arccos x\) for \(-1 \leqslant x \leqslant 1\). [3]

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

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

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.

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. Sketch, for \(0 \leq x \leq 2\pi\), the graph of \(y = \sin\left(x + \frac{\pi}{6}\right)\). [2]
2. Write down the exact coordinates of the points where the graph meets the coordinate axes. [3]
3. Solve, for \(0 \leq x \leq 2\pi\), the equation \(\sin\left(x + \frac{\pi}{6}\right) = 0.65\), giving your answers in radians to 2 decimal places. [5]

One of the diagrams below shows the graph of \(y = \arccos x\) Identify the graph of \(y = \arccos x\) Tick \((\checkmark)\) one box. [1 mark] \includegraphics{figure_4}

Given that \(\arcsin x = \frac{1}{6}\pi\), find \(x\). Find \(\arccos x\) in terms of \(\pi\).

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

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 \(\sin x\).

2 Sketch the curve with equation \(y = \tan x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
On the same diagram, sketch the curve with equation \(y = \tan ^ { - 1 } x\) for all \(x\).
State the geometrical relationship between the curves.