How do you use half-angle identities to find the exact values of sin (u/2) & cos (u/2) when sin(u)= - 4/5 and 3pi/2 <u<2pi?

1 Answer
Jan 25, 2016

#cos (u/2) = -2sqrt5/5#
#sin (u/2) = sqrt5/5#

Explanation:

#sin u = - 4/5# --> #cos ^2 u = 1 - sin^2 u = 1 - 16/25 = 9/25#
#cos u = 3/5# (cos u positive since u is in the Quadrant IV).
To find #sin (u/2)# and # cos (u/2)# apply the trig identities:
#cos 2a = 2cos^2 a - 1#
#cos 2a = 1 - 2sin^2 a#
a. Find #cos (u/2)#
#cos u = 3/5 = 2cos^2 (u/2) - 1#
#2cos^2 (u/2) = 1 + 3/5 = 8/5#
#cos^2 (u/2) = 8/10#
#cos (u/2) = sqrt8/sqrt10 = +- sqrt80/10 = +- 4sqrt5/10 = +- 2sqrt5/5#
#cos (u/2) = - 2sqrt5/5# (#cos (u/2)# negative, Quadrant II)
b. Find #sin (u/2)#
#3/5 = 1 - 2sin^2 (u/2)#
#2sin^2 (u/2) = 1 - 3/5 = 2/5#
#sin^2 (u/2) = 2/10 = 1/5#
#sin (u/2) = +- 1/sqrt5 = +- sqrt5/5#
#sin (u/2) = sqrt5/5# (#sin (u/2)# positive, Quadrant II)