Find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas for cot(u) = 3 in quadrant 2

half-angle-formulas

#1

Find the exact values of sin(u/2), cos(u/2), and tan(u/2) using the half-angle formulas for cot(u) = 3 in quadrant 2

Answer:

cot(u)=3,
tan(u)=1/3,
by applying formula
tan(u)= 2tan(u/2)/1-tan^2(u/2)=1/3
6tan(u/2)=1-tan^2(u/2)
let tan(u/2)=x
then,
6x=1-x^2
x^2+6x-1=0

Now by taking positive sign, tan(u/2) comes to be positive which is not possible as u/2 is in 2 quadrant. hence, only negative sign is considered.

taking negative sign,
x= (-6-√40)/2 = (-6-6.32)/2 = --12.32/2 = -6.16

tan(u/2) = -6.16

sec^2(u/2) = 1+tan^2(u/2)
sec^2(u/2) = 38.94
sec(u/2) = -6.24 (as u/2 is in 2 quadrant)

cos(u/2) = -0.1602

sin(u/2) = cos(u/2)tan(u/2) = (-0.1602)(-6.16)

sin(u/2) = 0.9868