Question

How do you use the half angle identity to find exact value of Sin (3pi/8)?

Answer 1

Find `sin ((3pi)/8)`

Ans: `sqrt(2 + sqrt2)/2`

Explanation:

This is the popular way to solve this kind of math question.
Call `sin a = sin ((3pi)/8)`
`cos 2a = cos ((6pi)/8) = cos ((3pi)/4) = -sqrt2/2`
Apply the trig identity: `cos 2a = 1 - 2sin^2 a`
`-sqrt2/2 = 1 - 2sin^2 a`
`2sin^2 a = 1 + sqrt2/2 = (2 + sqrt2)/2`
`sin^2 a = (2 + sqrt2)/4`
`sin a = sin ((3pi)/8) = +- sqrt(2 + sqrt2)/2.`
Since the arc `((3pi)/8)` is located in Quadrant I, its sin is positive, then
`sin ((3pi)/8) = sqrt(2 + sqrt2)/2`
Check by calculator.
`sin ((3pi)/8) = sin 67.5 deg = 0.92`
`sqrt(2 + sqrt2)/2 = (1.84)/2 = 0.92`. OK