Question

If $\tan \theta +\sec \theta =1.5$, then $\sin \theta =\frac{a}{13}$
Find $a$.

Answer 1

Given that $\tan \theta +\sec \theta =1.5$

We know that $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\sec \theta =\frac{1}{\cos \theta }$

So by putting these two we get

$2(1+\sin \theta )=3\cos \theta$

Squaring on both sides we get

$4(1+2\sin \theta +{\sin }^2\theta )=9(1-{\sin }^2\theta )$

$13{\sin }^2\theta +8\sin \theta -5=0$

solving which we get $\sin \theta =-1,$

$\sin \theta =-1,\frac{5}{13}$ this is not possible because the main equation is not satisfied
So we get $\sin \theta =\frac{5}{13}$ by comparing with $\frac{a}{13}$ we get

$\Rightarrow a=5$