Question

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

Answer 1

`\tan(\sin^( - 1)( (x)/(2) )) = \frac{x}{ \sqrt{4 - {x}^(2) } } \: \:where \: \: x \\e \pm2`

Step-by-step explanation

We want to find the exact value of

`\tan( \sin^( - 1)( (x)/(2) ) )`

Let

`y = \sin^( - 1)( (x)/(2) )`

This implies that

`\sin(y) = (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) = (opposite)/(adjacent)`

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

But

`y = \sin^( - 1)( (x)/(2) )`

This implies that,

`\tan(\sin^( - 1)( (x)/(2) )) = \frac{x}{ \sqrt{4 - {x}^(2) } } \: \:where \: \: x \\e \pm2`