Suppose \(0^\circ<\theta<90^\circ\) and \(\sec\theta+\tan\theta=5\). Using this, determine \(\sin\theta\).
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{12}{13}\)
The identity \(\sec^2\theta-\tan^2\theta=1\) factorises as \((\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1\). So the difference is the reciprocal of the given sum, and the two equations give \(\sec\theta\) and \(\tan\theta\).
\(\sec\theta-\tan\theta=\displaystyle\frac{1}{5}\)
Adding the two equations: \(2\sec\theta=5+\displaystyle\frac{1}{5}=\frac{26}{5}\), so \(\sec\theta=\displaystyle\frac{13}{5}\)
Subtracting: \(2\tan\theta=5-\displaystyle\frac{1}{5}=\frac{24}{5}\), so \(\tan\theta=\displaystyle\frac{12}{5}\)
\(\sin\theta=\tan\theta\times\cos\theta=\displaystyle\frac{12}{5}\times\frac{5}{13}=\frac{12}{13}\)