Question

If $\sin \theta =\frac{8}{17}$ and ${90}^o<\theta <{180}^o$, then the value of the expression $\frac{2\sin \theta +\cos \theta }{3\cos \theta +5\sin \theta }$ is :

• A. $\frac{1}{5}$
• B. $-\frac{1}{5}$
• C. $\frac{31}{85}$
• D. $-\frac{31}{85}$

Answer 1

Answer: (C)

$\frac{31}{85}$

Given, $\sin \theta =\frac{8}{17}=\frac{p}{h}$
$\therefore b=15,\cos \theta =\frac{b}{h}=\frac{15}{17}$
Therefore, $\frac{2\sin \theta +\cos \theta }{3\cos \theta +5\sin \theta }=\frac{2\left(\frac{8}{17}\right)+\frac{15}{17}}{3\left(\frac{15}{17}\right)+5\left(\frac{8}{17}\right)}$
$=\frac{\frac{16}{17}+\frac{15}{17}}{\frac{45}{17}+\frac{40}{17}}=\frac{\frac{31}{17}}{\frac{85}{17}}=\frac{31}{85}$