Find period or state transformations

9. \begin{figure}[h] \end{figure} 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 \(\cos \left( \frac { x } { 12 } \right) = k\).

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 \(\tan \theta\).

9 A curve has equation \(y = \sin ( a x )\), where \(a\) is a positive constant and \(x\) is in radians.

1. State the period of \(y = \sin ( a x )\), giving your answer in an exact form in terms of \(a\).
2. Given that \(x = \frac { 1 } { 5 } \pi\) and \(x = \frac { 2 } { 5 } \pi\) are the two smallest positive solutions of \(\sin ( a x ) = k\), where \(k\) is a positive constant, find the values of \(a\) and \(k\).
3. Given instead that \(\sin ( a x ) = \sqrt { 3 } \cos ( a x )\), find the two smallest positive solutions for \(x\), giving your answers in an exact form in terms of \(a\). \section*{END OF QUESTION PAPER}

5 Part of the graph of \(y = f ( x )\) is shown below. The graph is the image of \(y = \tan x ^ { \circ }\) after a stretch in the \(x\)-direction.
\includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-4_791_1022_1014_244}

1. Find the equation of the graph.
2. Write down the period of the function \(\mathrm { f } ( x )\).
3. In this question you must show detailed reasoning. Find all the roots of the equation \(\mathrm { f } ( x ) = 1\) for \(0 ^ { \circ } \leqslant x ^ { \circ } \leqslant 360 ^ { \circ }\).

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]
4. 
   \includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-12_675_1417_906_370}