Question

How do you find the exact values of sin (-17pi/12)+sin(pi/12) using the half angle formula?

Answer 1

Find exact values of `sin ((-17pi)/12) + sin (pi/12)`

Explanation:

Call `sin ((-17pi)/12) = sin x`
`cos 2x = cos ((-34pi)/12) = cos ((-10pi)/12) - 2pi) = cos ((-5pi)/6)`
`= cos ((5pi)/6) = -(sqrt3)/2`
`cos 2x = -(sqrt3)/2 = 1 - 2sin^2 x` --> `sin^2 x = (2 + sqrt3)/4`
-->`sin x = sin ((-17pi)/12) = sqrt(2 + sqrt3)/2`

Call `sin (pi/12) = sin y`
`cos 2y = cos (pi/6) = (sqrt3)/2 = 1 - 2sin^2 y`
`sin^2 y = (2 - sqrt3)/4 --> sin y = sqrt(2 - sqrt3)/2`

Answer: `sin x + sin y = sqrt(2 + sqrt3)/2 + sqrt(2 - sqrt3)/2`