Question

How do you use the angle sum identity to find the exact value of `sin285`?

Answer 1

`- sqrt(2 + sqrt3)/2`

Explanation:

Trig unit circle -->
sin 285 = sin (-75 + 360) = sin (-75)
Property of complementary arcs -->
sin (-75) = sin (-15 + 90) = - cos 15
Next, find (cos 15) by using trig identity:
`2cos^2 a = 1 + cos 2a`
`2cos^2 (15) = 1 + cos 30 = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 15 = (2 + sqrt3)/4`
`cos 15 = +- sqrt(2 + sqrt3)/2`
Since cos 15 is positive then take the positive value only.
Finally:
sin 285 = - cos 15 = `- sqrt(2 + sqrt3)/2`
Check by calculator:
sin 285 = - 0.965
`- sqrt(2 + sqrt3)/2 = 1.93/2 = -0.965.` OK