Question

How do you simplify using the half angle formula `sin(22^o 30')`?

Answer 1

`(sqrt(2-sqrt2))/2`

Explanation:

Using the `color(blue)"half angle formula for sin"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(sin(theta/2)=+-sqrt((1-costheta)/2))color(white)(2/2)|)))`

`"here "theta/2=22.5^@rArrtheta=45^@`

We only require the positive root since `22.5^@` is in the first quadrant where sine is positive.

`rArrsin(22.5^@)=sqrt((1-cos45^@)/2)`

`color(white)(XXXXXXX)=sqrt((1-1/sqrt2)/2)`

`color(white)(XXXXXXX)=sqrt((sqrt2-1)/(2sqrt2))`

`color(white)(xxxxxxxxx)=sqrt((2-sqrt2)/4`

`color(white)(xxxxxxxxx)=sqrt(2-sqrt2)/2`