Pick the value of \(\tan\theta\) for an acute angle \(\theta\) whose sine and cosine obey \(9\sin^2\theta+5\cos^2\theta=8\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\sqrt{3}\)
Write \(5\cos^2\theta\) as \(5(1-\sin^2\theta)\) using \(\sin^2\theta+\cos^2\theta=1\); the equation then has \(\sin^2\theta\) only. Since \(\theta\) is acute, \(\sin\theta\) is the positive root.
\(9\sin^2\theta+5-5\sin^2\theta=8\), so \(4\sin^2\theta=3\)
\(\sin\theta=\displaystyle\frac{\sqrt{3}}{2}\), so \(\theta=60^\circ\)
\(\tan\theta=\tan 60^\circ=\sqrt{3}\)