Pick the value of \(\sec\theta\) for an acute angle \(\theta\) whose sine and cosine obey \(7\sin^2\theta+3\cos^2\theta=4\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{2}{\sqrt{3}}\)
Write \(3\cos^2\theta\) as \(3(1-\sin^2\theta)\) 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.
\(7\sin^2\theta+3-3\sin^2\theta=4\), so \(4\sin^2\theta=1\)
\(\sin\theta=\displaystyle\frac{1}{2}\), so \(\theta=30^\circ\)
\(\sec\theta=\sec 30^\circ=\displaystyle\frac{2}{\sqrt{3}}\)