- #1
Gekko
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Homework Statement
Prove bijection between the regions
0<x<1, 0<y<1, 0<u, 0<v, u+v<pi/2
Homework Equations
x=sinu/cosv y = sinv/cosu
The Attempt at a Solution
We need to show that an inverse function exists to prove the bijection so obviously, (u,v) maps to one and only one (x,y) for the above. But what about the other way around? What is the best approach
Do we need to calculate:
u=arcsin(xcosv), v=arccos(sinu/x), u=arccos(sinv/y), v=arcsin(ycosu) and then look at each individually? Or could we divide one by the other and obtain tan(u)tan(v)=xy so that u=arctan(xy/tan(v) and v=arctan(xy/tan(u))?