Question

if $3\tan \theta =2$, find the value of $\frac{4\sin \theta -\cos \theta }{2\sin \theta +\cos \theta }$

• A. $\frac{3}{2}$
• B. $\frac{5}{7}$
• C. $\frac{3}{7}$
• D. $1$

Answer 1

The correct option is B $\frac{5}{7}$

Dividing the numerator and denominator of
$\frac{4\sin \theta -\cos \theta }{2\sin \theta +\cos \theta }$ by $\cos \theta$ , we get
$=\frac{4\sin \theta -\cos \theta }{2\sin \theta +\cos \theta }=\frac{4\frac{\sin \theta }{\cos \theta }-\frac{\cos \theta }{\cos \theta }}{2\frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\cos \theta }}$
$=\frac{4\tan \theta -1}{2\tan \theta +1}.......\left(i\right)$
Also, $3\tan \theta =2$ (given)
$\therefore \tan \theta =2/3$.
Substituting the value of $\tan \theta$ in (i) we get
$\frac{4\tan \theta -1}{2\tan \theta +1}=\frac{4\times \frac{2}{3}-1}{2\times \frac{2}{3}+1}=\frac{\frac{8}{3}-1}{\frac{4}{3}+1}$
$=\frac{5/3}{7/3}=\frac{5}{3}\times \frac{3}{7}=\frac{5}{7}$