Without using a calculator, show that \(\tan\dfrac\pi{12}=2-\sqrt3\).[2]
Show that \(\tan\left(\dfrac\pi4-\theta\right)=\dfrac{1-\tan\theta}{1+\tan\theta}\).[2]
A rotating stage turns through an angle \(\theta\) radians, where \(0\le\theta\le\dfrac\pi4\).
A sensor signal \(S\) is given by \(S=\dfrac{1-\tan\theta}{1+\tan\theta}\). The sensor is aligned when \(S\le2-\sqrt3\).
Using the results in (a) and (b), find the range of values of \(\theta\) for which the sensor is aligned.[3]
The rotating stage moves uniformly from \(\theta=0\) to \(\theta=\dfrac\pi4\) in 24 seconds.
The locking mechanism is activated only if the sensor remains aligned continuously for more than 7.5 seconds.
Determine whether the locking mechanism is activated. Justify your answer.[2]