Question

If 3 cot θ = 4, then $\frac{(5\mathrm{sin\theta }+3\mathrm{cos\theta })}{(5\mathrm{sin\theta }-3\mathrm{cos\theta })}$ = ?
(a) $\frac{1}{3}$
(b) 3
(c) $\frac{1}{9}$
(d) 9

Answer 1

(d) 9

We have $\frac{\left(5\sin \theta +3\cos \theta \right)}{\left(5\sin \theta -3\cos \theta \right)}$.

Dividing the numerator and denominator of the given expression by sin θ, we get:
$\frac{{\displaystyle \frac{1}{\sin \theta }}\left(5\sin \theta +3\cos \theta \right)}{{\displaystyle \frac{1}{\sin \theta }}\left(5\sin \theta -3\cos \theta \right)}$

= $\frac{5+3cot\theta }{5-3\mathrm{co}t\theta }$

= $\frac{5+4}{5-4}$ = 9 [∵ 3 cot θ = 4]