Question

How do you use the half angle identity to find exact value of sin195?

Answer 1

Find sin 195

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

Explanation:

Call sin 195 = sin t
`cos 2t = cos 390 = cos (30 + 360) = cos 30 = sqrt3/2`
Apply the trig identity: `cos 2t = 1 - 2sin^2 t`
`cos 2t = sqrt3/2 = 1 - 2sin^2 t`
`2sin^2 t = 1 - sqrt3/2 = (2 -sqrt3)/2`
`sin^2 t = (2 - sqrt3)/4`
`sin 195 = sin t = +- sqrt(2 - sqrt3)/2`
Since the arc 195 is located in Quadrant III, its sin is negative. Then,
`sin 195 = - sqrt(2 - sqrt3)/2`
Check by calculator.
sin 195 = -0.26
`-sqrt(2 - sqrt3)/2 = -0,52/2 = -0.26`. OK