Question

If $\sin \theta =\frac{12}{13}$, then the value of $\frac{2\cos \theta +3\tan \theta }{\sin \theta +\tan \theta \sin \theta }$

• A. $\frac{12}{5}$
• B. $\frac{5}{13}$
• C. $\frac{259}{102}$
• D. $\frac{259}{65}$

Answer 1

The correct option is C $\frac{259}{102}$

$sin\theta =\frac{12}{13}=\frac{P}{H}$
$P=12,H=13$
$\Rightarrow P^2+B^2=H^{2}B=\sqrt{H^2-P^2}=\sqrt{{13}^2-{12}^2}=5$
Hence
$cos\theta =\frac{B}{H}=\frac{5}{13}$
$tan\theta =\frac{P}{B}=\frac{12}{5}$
Substituting in the given equation, we get
$\frac{2\frac{5}{13}+3\frac{12}{5}}{\frac{12}{13}+\frac{12\times 12}{5\times 13}}$
$=\frac{50+468}{144+60}$
$=\frac{518}{204}$
$=\frac{259}{102}$