How do you evaluate tan(sin^-1(-5/11)) without a calculator?

Oct 11, 2016

$- - \frac{5}{24} \sqrt{6}$.

Explanation:

Let $a = {\sin}^{- 1} \left(- \frac{5}{11}\right) \in {Q}_{4}$, for the principal value.

Then, $\sin a = - \frac{5}{11} , \cos a = \sqrt{1 - {\left(- \frac{5}{11}\right)}^{2}} = \frac{4}{11} \sqrt{6}$

The given expression is

$\tan a = \sin \frac{a}{\cos} a = - \frac{5}{4 \sqrt{6}} = - \frac{5}{24} \sqrt{6}$

Of course, the general value $a \in {Q}_{3}$ also. wherein tana is

positive and cos a is negative,

So, ignoring the principal-value-convention for inverse sine, the

$= \pm \frac{5}{24} \sqrt{6}$.