Sketch single trig graph

6

1. Sketch the graph of the curve \(y = 3 \sin x\), for \(- \pi \leqslant x \leqslant \pi\). The straight line \(y = k x\), where \(k\) is a constant, passes through the maximum point of this curve for \(- \pi \leqslant x \leqslant \pi\).
2. Find the value of \(k\) in terms of \(\pi\).
3. State the coordinates of the other point, apart from the origin, where the line and the curve intersect.

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 \(k\).

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

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

5

1. Fig. 5 shows the graph of a sine function. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{50ebbd77-39da-4a48-993a-bcf99ada9dcd-3_534_1154_312_450} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure} State the equation of this curve.
2. Sketch the graph of \(y = \sin x - 3\) for \(0 ^ { \circ } \leqslant x \leqslant 450 ^ { \circ }\).

4. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180$$

1. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\) axis.
2. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
3. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\)
   [0pt] [P1 June 2002 Question 5]

4. (a) (i) Sketch the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
(ii) Write down the coordinates of the turning points of the curve in this interval.
(b) Find all values of \(x\) in the interval \(- 180 \leq x \leq 180\) for which $$\sin ( x - 30 ) ^ { \circ } = 0.35$$ giving your answers to 1 decimal place.

6

1. Sketch, on the axes below, the curve with equation \(y = \sin ^ { - 1 } ( 3 x )\), where \(y\) is in radians. State the exact values of the coordinates of the end points of the graph.
2. Given that \(x = \frac { 1 } { 3 } \sin y\), write down \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) and hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). \section*{Answer space for question 6}
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-14_839_1451_813_324}

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}