Find the inverse of following function

[sit.think.solve]

How do you find the inverse of following function?

p(x)=(cos2pix,sin2pix) where pi means 3.14etc NOT p times i.

 

[HallsofIvy]

This is a function from R to R²? I.e. p(t)= (cos(2\pi x), sin(2\pi x). Then it maps the half open interval [0, 1) in R onto the unit circle in R². At t=1 you start around the circle again so the function is not one-to-one and you do not have an inverse outside that interval. The range of that function is, of course, {(x,y)|x²+ y²= 1} and that must be the domain of the inverse function. On that domain, p^{-1}(x,y)= arccos(x)/(2\pi) if y>= 0, 1- arcos(x)/(2\pi) if y< 0.