Calculate intersection coordinates algebraically

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\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 \(B\).

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\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 \(\alpha\),
   (a) the next smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\),
   (b) the smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = - k\).
2. 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 \(y = \sin \frac { 1 } { 3 } x\).
3. 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}

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1. Sketch the graphs of \(y = 4 \cos x\) and \(y = 2 \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) on the same axes.
2. Find the exact coordinates of the point of intersection of these graphs, giving your answer in the form (arctan \(a , k \sqrt { b }\) ), where \(a\) and \(b\) are integers and \(k\) is rational.
3. A student argues that without the condition \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\) all the points of intersection of the graphs would occur at intervals of \(360 ^ { \circ }\) because both \(\sin x\) and \(\cos x\) are periodic functions with this period. Comment on the validity of the student's argument.

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1. The graph of \(y = 3 \sin ^ { 2 } \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\) is shown in Fig. 6.
   On the copy of Fig. 6 in the Printed Answer Booklet, sketch the graph of \(y = 2 \cos \theta\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-05_818_1507_571_351} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. In this question you must show detailed reasoning. Determine the values of \(\theta , 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\), for which the two graphs cross.

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

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.
（a）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 \(T\) ．
（b）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 \(0 < \alpha < \pi\) ．
（c）Show that \(\alpha = \arccos \left( \frac { \pi } { 4 } \right)\) ．
（d）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 \(\tan \beta\) ．
（e）Show that \(\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)\) ．
（f）Find，in terms of \(\beta\) ，the obtuse angle between the tangent to \(C _ { 1 }\) at \(S\) and the tangent to \(C _ { 2 }\) at \(S\) ．

2 In this question you must show detailed reasoning. Fig. 2 shows the graphs of \(y = 4 \sin x ^ { \circ }\) and \(y = 3 \cos x ^ { \circ }\) for \(0 \leqslant x \leqslant 360\). \begin{figure}[h] \end{figure} Find the \(x\)-coordinates of the two points of intersection, giving your answers correct to 1 decimal place.

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(T\) with equation \(y = \cos 2 x\) and the circle \(C _ { 1 }\) that touches \(T\) at \(\left( \frac { \pi } { 4 } , 0 \right)\) and \(\left( \frac { 3 \pi } { 4 } , 0 \right)\) ．
（a）Find the radius of \(C _ { 1 }\) \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of \(T\) and part of a circle \(C _ { 2 }\) that touches \(T\) at the point \(P\) with coordinates \(\left( \frac { \pi } { 2 } , - 1 \right)\) ．For values of \(x\) close to \(\frac { \pi } { 2 }\) the curve \(T\) lies inside \(C _ { 2 }\) as shown in Figure 3.
（b）Without doing any calculation，explain why the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(C _ { 2 }\) at \(P\) is less than the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(T\) at \(P\) ． The radius of \(C _ { 2 }\) is \(r\) ．
（c）Use the result from part（b）to find a value of \(k\) such that \(r > k\) ． Given that \(C _ { 2 }\) cuts \(T\) at the point \(( 0,1 )\) ，
（d）find the value of \(r\) ．