If \(7\tan\theta=5\), with \(0^\circ<\theta<90^\circ\), what is the value of the expression \(\displaystyle\frac{3\sin\theta+2\cos\theta}{5\sin\theta-\cos\theta}\)?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{29}{18}\)
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.
\(\tan\theta=\displaystyle\frac{5}{7}\)
Expression \(=\displaystyle\frac{3\tan\theta+2}{5\tan\theta-1}\)
Top: \(3\times\displaystyle\frac{5}{7}+2=\frac{15}{7}+\frac{14}{7}=\frac{29}{7}\); bottom: \(5\times\displaystyle\frac{5}{7}-1=\frac{25}{7}-\frac{7}{7}=\frac{18}{7}\)
Value \(=\displaystyle\frac{29}{7}\div\frac{18}{7}=\frac{29}{18}\)