csc θ = 3, θ lies in quadrant I. Find cos (θ/2)

Answer 1

csc(theta) = 3 = 1/sin(theta)

sin(theta) = 1/3

cos(theta) = sqr(3^2-1^2)/3 sqr(8)/3 = 2sqr2/3 = 2sqr2/3

cos(theta) = (2/3)sqr2 = about 0.9428

theta = about 19.47

theta/2 = about 9.7356

cos(theta/2) = about 0.9856

or for the "exact" solution use the half angle formula

cos(theta/2) = square root of (1+cos(theta))/2

cos(theta/2) = exactly sqr((1+2sqr2/3)/2)

= about 0.9856