half angle identity?

Question

half angle identity 5pi/12

ne 1 know how to work this out?

Answer 1

Are you asking for trig values for 5π/12 using half-angle identities? If so, the identities we need are

sin²(x/2) = (1 - cos(x))/2
cos²(x/2) = (1 + cos(x))/2

5π/12 is in quadrant I, so both its sine and its cosine are positive.

So

sin(5π/12) = sqrt[(1 - cos(5π/6))/2]
cos(5π/12) = sqrt[(1 + cos(5π/6))/2]

5π/6 has reference angle of π/6 and is in quadrant II so

cos(5π/6) = -cos(π/6) = -sqrt(3)/2

So

sin(5π/12) = sqrt[(1 - {-sqrt(3)/2})/2] = sqrt[2 + sqrt(3)]/2
cos(5π/12) = sqrt[(1 + {-sqrt(3)/2})/2] = sqrt[2 - sqrt(3)]/2

Answer 2

(sin ^2( 6x))cos^2 (6x)) = a million/4( sin ^2 (12x)) employing sin 2a= 2 sin a cos a so sin a cos a= sin 2a/2 = a million/4( a million- cos 24x)/2 as cos2a = a million= 2 sin ^2 2a = a million/8 - (cos 24x)/8 and integrating we get x/8 - (sin 24x)/192+ C