If \(5\sin\theta=12\cos\theta\) and \(0^\circ<\theta<90^\circ\), find \(\sec\theta\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{13}{5}\)
Dividing both sides by \(\cos\theta\) turns the equation into a value of \(\tan\theta\); then \(\sec^2\theta=1+\tan^2\theta\) gives the secant.
\(\tan\theta=\displaystyle\frac{\sin\theta}{\cos\theta}=\frac{12}{5}\)
\(\sec^2\theta=1+\displaystyle\frac{144}{25}=\frac{169}{25}\)
\(\theta\) is acute, so \(\sec\theta\) is positive: \(\sec\theta=\displaystyle\frac{13}{5}\)