Question

How do you use the half angle identity to find exact value of sin 15 degrees?

Answer 1

Find sin (15)

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

Explanation:

Use the trig identity: `cos 2a = 1 - 2sin^2 a`
`cos (30) = sqrt3/2 = 1 - 2sin^2 (15)`
`2sin^2 (15) = 1 - sqrt3/2 = (2 - sqrt3)/2`
`sin^2 (15) = (2 - sqrt3)/4`
`sin (15) = +- sqrt(2 - sqrt3)/2.`
Since arc (15) deg is in Quadrant I, its sin is positive. Then,
`sin (15) = sqrt(2 - sqrt3)/2.`

Check by calculator.
sin 15 = 0.26
`sqrt(2 + sqrt3)/2 = 0.52/2 = 0.26`. OK