Question

Find the exact value by using a half-angle identity.

sin(5 pi /12 )

Answer 1

Can anybody help

Answer 2

Use Θ = 5π/6. Then the half angle Θ/2 = 5π/12. Fortunately, we know what the cosine of 5π/6 is (it has reference angle π/6 and is in the second quadrant.

cos(5π/6) = -√(3)/2.

Now, 5π/12 is in quadrant I, so the sine of it is positive. Using the half angle ID

sin(5π/12) = √[(1 - cos(5π/6))/2] = √[(1 + √(3)/2)/2] = √[(2 + √(3))/4] = √[2 + √(3)]/2.