Let sin\beta =\frac{2\sqrt[]{2} }{5} \\ and (\pi )/(2) \leq\beta \leq \pi \\. Determine the exact value of sin((\beta )/(2) )

Question

Let
`sin\beta =\frac{2\sqrt[]{2} }{5} \\` and
`(\pi )/(2) \leq\beta \leq \pi \\`.

Determine the exact value of
`sin((\beta )/(2) )`

Answer 1

Since beta is in the first quadrant, the final answer will be positive.

To find cos(beta) so we can use the half angle identity, we can substitute into the Pythagorean identity. Doing so gives us that

`\sin( \beta ) = ( √(17) )/(5)`

So, this means that

`\sin( ( \beta )/(2) ) = \sqrt{ (1 - ( √(17) )/(2) )/(2) } = \sqrt{ (2 - √(17) )/(4) } = \frac{ \sqrt{2 - √(17) } }{2}`