Question

How do you find the exact values of sin 105 using the half angle formula?

Answer 1

Find sin 105.
Answer: `sin (105) = +- sqrt(2 + sqrt3)/2`

Explanation:

Call sin 105 = sin t
`cos (210) = cos (2t) = cos (30 + 180) = - cos (30) = (-sqrt3)/2`
Apply the trig identity: `cos 2a = 1 - 2sin^2 a.`
`cos 210 = (-sqrt3)/2 = 1 - 2sin^2 t`
`2sin^2 t = 1 + sqrt3/2 = (2 + sqrt3)/2`
`sin^2 t = (2 + sqrt3)/4`-->

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