Question

If $\tan \theta +\sec \theta =1.5cos\theta$, find $\sin \theta ,\tan \theta$, and $\sec \theta$.

Answer 1

$\Rightarrow \frac{\sin \theta +1}{\cos \theta }=\frac{3}{2}$

$\Rightarrow 2(\sin \theta +1)=3(1-{\sin }^2\theta )$

$\Rightarrow 3{\sin }^2\theta +2\sin \theta -1=0$

$\Rightarrow 3{\sin }^2\theta +3\sin \theta -\sin \theta -1=0$

$\Rightarrow 3\sin \theta (\sin \theta +1)-1(\sin \theta +1)=0$

$\Rightarrow \sin \theta =-1$ or $\sin \theta =1/3$

$\Rightarrow \tan \theta =\infty$ or $\tan \theta =1/2\sqrt{2}$

$\Rightarrow \sec \theta =\infty$ or $\sec \theta =1/2\sqrt{2}$.