# If tantheta>0 and csctheta=2, find all the trigonometrical ratios of theta?

Oct 19, 2017

$\sin \theta = \frac{1}{2}$, $\csc \theta = 2$, $\cos \theta = \frac{\sqrt{3}}{2}$, $\sec \theta = \frac{2}{\sqrt{3}}$, $\tan \theta = \frac{1}{\sqrt{3}}$ and $\cot \theta = \sqrt{3}$

#### Explanation:

$\tan \theta > 0$ mean $\theta$ is either in $Q 1$ or in $Q 3$

As $\csc \theta = 2$, $\theta$ is in $Q 1$ only.

As $\csc \theta = 2$, $\sin \theta = \frac{1}{2} = \sin {30}^{\circ}$

Hence $\theta = {30}^{\circ}$

So,
$\cos {30}^{\circ} = \frac{\sqrt{3}}{2} , \sec {30}^{\circ} = \frac{2}{\sqrt{3}} , \tan {30}^{\circ} = \frac{1}{\sqrt{3}} = 0.57 \text{ which is greater than } 0 , \cot {30}^{\circ} = \sqrt{3} , \sin {30}^{\circ} = \frac{1}{2} \mathmr{and} \csc {30}^{\circ} = 2$