If \(\cos\left(\displaystyle\frac{\pi}{2}-2\theta\right)=\frac{1}{2}\), then what is the value of \(\tan 3\theta\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 1
An angle and its complement are linked by \(\cos(90^\circ-A)=\sin A\), and \(\frac{\pi}{2}\) is \(90^\circ\). So the condition gives \(\sin 2\theta\), and from it \(\theta\).
\(\cos\left(\frac{\pi}{2}-2\theta\right)=\sin 2\theta=\frac{1}{2}\)
\(\sin 30^\circ=\frac{1}{2}\), so \(2\theta=30^\circ\) and \(\theta=15^\circ\)
\(\tan 3\theta=\tan 45^\circ=1\)
The other angle with sine \(\frac{1}{2}\) is \(150^\circ\); then \(\theta=75^\circ\) and \(\tan 225^\circ=1\) as well. Adding a multiple of \(180^\circ\) to \(\theta\) changes \(3\theta\) by a multiple of \(180^\circ\), so \(\tan 3\theta=1\) for every solution.