9 A robotic arm which is attached to a flat surface at the origin \(O\), is used to draw a graphic design.
The arm is made from two rods \(O P\) and \(P Q\), each of length \(d\), which are joined at \(P\).
A pen is attached to the arm at \(Q\).
The coordinates of the pen are controlled by adjusting the angle \(O P Q\) and the angle \(\theta\) between \(O P\) and the \(x\)-axis.
For this particular design the pen is made to move so that the two angles are always equal to each other with \(0 \leq \theta \leq \frac { \pi } { 2 }\) as shown in Figure 2.
\begin{figure}[h]
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\caption{Figure 2}
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\end{figure}
9
- Show that the \(x\)-coordinate of the pen can be modelled by the equation
$$x = d \left( \cos \theta + \sin \left( 2 \theta - \frac { \pi } { 2 } \right) \right)$$
9
- Hence, show that
$$x = d \left( 1 + \cos \theta - 2 \cos ^ { 2 } \theta \right)$$
9
- It can be shown that
$$x = \frac { 9 d } { 8 } - d \left( \cos \theta - \frac { 1 } { 4 } \right) ^ { 2 }$$
State the greatest possible value of \(x\) and the corresponding value of \(\cos \theta\)
9
- Figure 3 below shows the arm when the \(x\)-coordinate is at its greatest possible value.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-14_570_773_456_630}
\end{figure}
Find, in terms of \(d\), the exact distance \(O Q\).
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-15_2488_1716_219_153}