How do you evaluate tan(csc^-1(-2)) without a calculator?

Aug 19, 2016

$\tan \left({\csc}^{-} 1 \left(\frac{\sqrt{7}}{2}\right)\right) = \frac{1}{\sqrt{3}}$

Explanation:

Let $\theta = {\csc}^{-} 1 \left(- 2\right)$

or

$\csc \theta = - 2$

or

$\frac{1}{\sin} \theta = - 2$

or

$\sin \theta = - \frac{1}{2}$

or

$\theta = {\sin}^{-} 1 \left(- \frac{1}{2}\right)$

Hence

$\frac{p}{h} = \frac{1}{2}$ (Since $\theta$ lies in third quadrant; hence $-$ sign not taken in consideration)

$\cos \theta = \frac{b}{h}$

or

$\cos \theta = \frac{\sqrt{{h}^{2} - {p}^{2}}}{h}$

or

$\cos \theta = \frac{\sqrt{\left({2}^{2} - {1}^{2}\right)}}{2}$

or

$\cos \theta = \frac{\sqrt{4 - 1}}{2}$

or

$\cos \theta = \frac{\sqrt{3}}{2}$

Therefore
$\tan \theta = \sin \frac{\theta}{\cos} \theta$

or

tan theta=(1/2)/((sqrt3)/2

or

$\tan \theta = \frac{1}{\sqrt{3}}$

or

$\tan \left({\csc}^{-} 1 \left(\frac{\sqrt{7}}{2}\right)\right) = \frac{1}{\sqrt{3}}$