Question

If $2\tan \theta +\sec \theta =5$, find $\sin \theta$.

Answer 1

Given $2\tan \theta +\text{sec}\theta =5$

$⟹\frac{2\sin \theta +1}{\cos \theta }=5$

$⟹(2\sin \theta +1{)}^2=(5\cos \theta {)}^2$

$⟹4{\sin }^2\theta +1+4\sin \theta =25{\cos }^2\theta$

$⟹4{\sin }^2\theta +4\sin \theta +1=25-25{\sin }^2\theta$

$⟹29{\sin }^2\theta +4\sin \theta -24=0$

$⟹\sin \theta =\frac{-4\pm \sqrt{16+4\left(29\right)\left(24\right)}}{2\times 29}=\frac{-2\pm 10\sqrt{7}}{29}$