Question

Find the exact value of the following expression (without using a calculator): tan(Sin^-1 x/2)

Answer 1

ANSWER

$\tan(\sin^{ - 1}( \frac{x}{2} )) = \frac{x}{ \sqrt{4 - {x}^{2} } } \: \:where \: \: x \ne \pm2$

EXPLANATION

We want to find the exact value of

$\tan( \sin^{ - 1}( \frac{x}{2} ) )$

Let

$y = \sin^{ - 1}( \frac{x}{2} )$

This implies that

$\sin(y) = \frac{x}{2}$

This implies that,

The opposite is x units and the hypotenuse is 2 units.

The adjacent side is found using Pythagoras Theorem.

${a}^{2} + {x}^{2} = {2}^{2}$

${a}^{2} + {x}^{2} = 4$

${a}^{2} = 4 - {x}^{2}$

$a= \sqrt{4 - {x}^{2}}$

This implies that,

$\tan(y) = \frac{opposite}{adjacent}$

$\tan(y) = \frac{x}{ \sqrt{4 - {x}^{2} } }$

But

$y = \sin^{ - 1}( \frac{x}{2} )$

This implies that,

$\tan(\sin^{ - 1}( \frac{x}{2} )) = \frac{x}{ \sqrt{4 - {x}^{2} } } \: \:where \: \: x \ne \pm2$