if cos(theta)=1/4 and theta is in quadrant IV, find tan(theta/2)

If cosθ = 1/4    we need to find the sinθ

So we have

cos^2θ + sin^2θ  = 1

[1/4]^2 + sin^2θ  = 1

1/16  + sin^2θ = 1

sin^2θ   = 1 -1/16   = 15/16     take the square root of both sides

sinθ  = ± √15 / 4       since θ  is in quad IV....the sine will be negative here

And tan(θ/2) = [ sinθ ] / [1  + cosθ]  = [ - √15 / 4] /  [1 + 1/4]  =

[ - √15 / 4] / [ 5/4]   =    [ - √15 / 4]  * [ 4 / 5]   = [ -4√15] / [20]   =  -√15 / 5

This is the exact value.....!!!

Note...since θ  is in the 4th quadrant, θ/2 will lie in the second quadrant.....and the tangent will be negative there.......just as we found...!!!!