For how many values of x such that 0<x<360 do we have cos(x) = 0.1? (x is in radians)

You will have to use a calculator for this one.

if cosx = .1
then x = 84.26° correct to 2 decimals
but cosine is positive in I or IV
so x = 360-84.26 = 275.74°

x = appr 84.26 or 275.74