Question

If $5\cos \theta =6\sin \theta$; evaluate:
$\frac{12\sin \theta -3\cos \theta }{12\sin \theta +3\cos \theta }$

Answer 1

$5cos\theta =6sin\theta \Rightarrow \frac{cos\theta }{sin\theta }=\frac{6}{5}$

$\frac{12sin\theta -3cos\theta }{12sin\theta +3cos\theta }=\frac{3sin\theta }{3sin\theta }\left(\frac{4-\frac{cos\theta }{sin\theta }}{4+\frac{cos\theta }{sin\theta }}\right)$

$=\frac{4-\frac{6}{5}}{4+\frac{6}{5}}=\frac{20-6}{20+6}=\frac{14}{26}=\frac{7}{13}$