Question

How do you use the half angle identity to find exact value of sin 75 degrees?

Answer 1

Find sin (75)

Ans: `sqrt(2 + sqrt3)/2`

Explanation:

Call sin 75 = sin t
Apply the trig identity: `cos 2t = 1 - 2sin^2 t`
`cos 2t = cos 150 = -sqrt3/2 = 1 - 2sin^2 t`
`2sin^2 t = 1 + sqrt3/2`
`sin^2 t = (2 + sqrt3)/4`
`sin t = +- sqrt(2 + sqrt3)/2`
Since the arc 75 is in Quadrant I, its sin is positive, therefor

`sin t = sin 75 = sqrt(2 + sqrt3)/2`

Check by calculator.
sin 75 = 0.97
`sqrt(2 + sqrt3)/2 = (sqrt3.73)/2 = (1.93)/2 = 0.97`. OK