Let \(\theta\) be an acute angle such that \(3\tan\theta=2\). Evaluate \(\displaystyle\frac{5\sin\theta-3\cos\theta}{5\sin\theta+2\cos\theta}\).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{1}{16}\)
Every term has \(\sin\theta\) or \(\cos\theta\) to the first power, so dividing the top and the bottom by \(\cos\theta\) turns the expression into one in \(\tan\theta\) only.
\(\tan\theta=\displaystyle\frac{2}{3}\)
Expression \(=\displaystyle\frac{5\tan\theta-3}{5\tan\theta+2}\)
\(5\tan\theta=\displaystyle\frac{10}{3}\), so the top is \(\displaystyle\frac{10}{3}-3=\frac{1}{3}\) and the bottom is \(\displaystyle\frac{10}{3}+2=\frac{16}{3}\)
Value \(=\displaystyle\frac{1}{3}\div\frac{16}{3}=\frac{1}{16}\)