Question

If $5\cos \theta -12\sin \theta =0$, find $\frac{\sin \theta +\cos \theta }{2\cos \theta -\sin \theta }$

Answer 1

$5\cos \theta -12\sin \theta =0$

$\Rightarrow 5\cos \theta =12\sin \theta$

$\Rightarrow \cot \theta =\frac{12}{5}$

$\Rightarrow \frac{\sin \theta +\cos \theta }{2\cos \theta -\sin \theta }=\frac{\sin \theta (1+\cot \theta )}{\sin \theta (2\cot \theta -1)}$

$=\frac{1+\frac{12}{5}}{2\times \frac{12}{5}-1}$

$=\frac{\frac{17}{5}}{\frac{24}{5}-1}$

$=\frac{\frac{17}{5}}{\frac{24-5}{5}}$

$=\frac{17}{5}\times \frac{5}{19}$

$=\frac{17}{19}$

Hence, the answer is $\frac{17}{19}.$