\(P\) is a point on the circumference of a circle \(X\) of diameter \(\quantity{3}{in.}\) An arc of another circle \(Y\) is drawn with centre \(P\), intersecting \(X\) at \(Q\) and \(R\). Express the lengths of \(PQ\) and the minor arc \(QR\) of \(Y\) in terms of \(\theta\), where \(\theta\) is the angle, in radians, between \(PQ\) and the diameter of \(X\) through \(P\).

We ought to begin by drawing the configuration described in the question. How can we find the length of \(PQ\)? Could we try adding to the configuration?

If we let \(P'\) denote the other end of the diameter of \(X\) that passes through \(P\), what can we say about \(\triangle P'QP\)?

Do we know anything about \(\angle P'QP\)?

Prove that the length of the arc \(QR\) has a stationary value when \(\theta = \cot \theta\), and that this value is a maximum.

When checking that a value is a maximum of a function, do we have to calculate the second derivative of the function?