Find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas

Question

 Find the exact values of sin(u/2),

 cos(u/2),

 and tan(u/2)

 using the half-angle formulas

tan(u) = −3/4,  3𝜋/2 < u < 2𝜋

Answer 1

tanu = -3/4

sinu = -3/5

cosu = 4/5

tanu = -1/3

sinu = sqr10/10

cosu = -3sqr10/10

u = tan^-1(-3/4) with u in the 4th quadrant = about 360-36.9 =323.1 degrees

where all sines<0 and cosines>0

u/2 is in 2nd quadrant= 323.1/2 = 161.6 degrees

where all sines>0 and cosines<0

sin(u/2 = sin161.6 = .32

sin(u/2) = sqr[(1-cosu)/2]= sqr(1-4/5)/2=sqr(1/10) = 1/sqr10 = sqr10/10 = about 3.16/10 = about .32

cos(u/2) = -sqr(1+cosu)/2=-sqr(1+4/5)/2 = -sqr(9/10)=-3/sqr10 = -3sqr10/10= about cos161.2 = -.95

tan(u/2) = sin(u/2)/cos(u/2) =about .32/.95= -.33

= (sqr10/10)/(-3sqr10/10) = -1/3