Sketch and solve graphically

10

1. Solve the equation \(2 \cos x + 3 \sin x = 0\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Sketch, on the same diagram, the graphs of \(y = 2 \cos x\) and \(y = - 3 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. Use your answers to parts (i) and (ii) to find the set of values of \(x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\) for which \(2 \cos x + 3 \sin x > 0\).

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.

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}

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

3 Sketch the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.2\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

6

1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve the equation \(2 \sin \theta = - 1\) for \(0 \leqslant \theta \leqslant 2 \pi\). Give your answers in the form \(k \pi\).

7

1. Sketch the graph of \(y = \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   On the same axes, sketch the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). Label each graph clearly.
2. Solve the equation \(\cos 2 x = 0.5\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5

1. Sketch the graph of \(y = \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   On the same axes, sketch the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). Label each graph clearly.
2. Solve the equation \(\cos 2 x = 0.5\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

7 Sketch the curve \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.68\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

8

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

9 Sketch the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
Solve the equation \(\sin x = - 0.2\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

10

1. Sketch the graph of \(y = \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   On the same axes, sketch the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). Label each graph clearly.
2. Solve the equation \(\cos 2 x = 0.5\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

12

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

9

1. \includegraphics[max width=\textwidth, alt={}, center]{4d03f4e3-ae6c-4a0d-ae4d-d89258d2919a-4_362_979_1505_625} The diagram shows part of the curve \(y = \cos 2 x\), where \(x\) is in radians. The point \(A\) is the minimum point of this part of the curve.
   1. State the period of \(y = \cos 2 x\).
   2. State the coordinates of \(A\).
   3. Solve the inequality \(\cos 2 x \leqslant 0.5\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.
2. Solve the equation \(\cos 2 x = \sqrt { 3 } \sin 2 x\) for \(0 \leqslant x \leqslant \pi\), giving your answers exactly.

9. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve with equation \(y = 3 \cos x ^ { \circ }\).
The point \(P ( c , d )\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.

1. State the value of \(c\) and the value of \(d\).
2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3 \cos x ^ { \circ }\) to the curve with equation
   1. \(y = 3 \cos \left( \frac { x ^ { \circ } } { 4 } \right)\)
   2. \(y = 3 \cos ( x - 36 ) ^ { \circ }\)
3. Solve, for \(450 ^ { \circ } \leqslant \theta < 720 ^ { \circ }\), $$3 \cos \theta = 8 \tan \theta$$ giving your solution to one decimal place.
   In part (c) you must show all stages of your working.
   Solutions relying entirely on calculator technology are not acceptable.

7 The diagram shows the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-4_518_906_1098_552}

1. Write down the coordinates of the points \(A , B\) and \(C\) marked on the diagram.
2. Describe the single geometrical transformation by which the curve with equation \(y = \cos 2 x\) can be obtained from the curve with equation \(y = \cos x\).
3. Solve the equation $$\cos 2 x = 0.37$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). (No credit will be given for simply reading values from a graph.)
   (5 marks)

7

1. Solve the equation \(\sin x = 0.8\) 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 the curve \(y = \sin x , 0 \leqslant x \leqslant 2 \pi\) and the lines \(y = k\) and \(y = - k\). \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-5_497_780_552_689} The line \(y = k\) intersects the curve at the points \(P\) and \(Q\), and the line \(y = - k\) intersects the curve at the points \(R\) and \(S\). 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 the point \(Q\) in terms of \(\pi\) and \(\alpha\).
   3. Find the length of \(R S\) in terms of \(\pi\) and \(\alpha\), giving your answer in its simplest form.
3. Sketch the graph of \(y = \sin 2 x\) for \(0 \leqslant x \leqslant 2 \pi\), indicating the coordinates of points where the graph intersects the \(x\)-axis and the coordinates of any maximum points.

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 \(\pi\).