Given tan 𝜃 = 5/4 and 𝜋 < 𝜃 < 3𝜋/2 Find the exact value of cos 𝜃/2

Answer 1

tanx = 5/4 = opposite side/adjacent side, hypotenuse = h = sqr(25+16) =sqr41, x is in 1st or 3rd quadrants, adjacent side is either >0 or <0

cosx = adjacent side/hypotenuse = 4/sqr41 or -4/sqr41

use the double angle formula

cos2x = 2cos^2x -1 = 2(4/sqr41)^2 -1 32/(41)-1 =-9/41

x=51.34, 2x = 101.68 cos2x = -.2195

-9/41 =-.2195

exact value of cos2x = -9/41