Find the exact values of sin(𝜃/2), cos(𝜃/2), and tan(𝜃/2) for the given condition.

Question

Find the exact values of sin(𝜃/2), cos(𝜃/2), and tan(𝜃/2) for the given condition.

csc 𝜃 = − 5/4; −90° < 𝜃 < 0°

sin(𝜃/2)

=

cos(𝜃/2)

=

tan(𝜃/2)

=

Answer 1

theta is in quadrant IV

sin(theta) = 1/csc(theta) = -4/5

It's a 3-4 5 right triangle

cos(theta) = 3/5

sec(theta) = 5/3

tan(theta) = -4/3

cot(theta) = -3/4

use the half angle identities to find tan(theta/2), sin(theta/2) and cos(theta/2)

sin(theta/2) = -sqr((1-cos(theta)/2) =-sqr((1-3/5)/2) = -sqr(1/5) =-1/sqr5

cos(theta/2) =sqr((1+cos(theta)/2) = sqr((1+3/5)/2) = sqr(4/5) = 2/sqr5

tan(theta/2) = -sqr((1-cos(theta)/(1+cos(theta)) = - sqr(1/4) = -1/2