Question

Show that the locus in the complex plane of all points satisfying `cosv + isinv` where `v in [0,2pi]` is a unit circle?

Answer 1

Suppose we have a point `z` in the complex plane such that

`z = cosv + isinv \ \` where `v in [0,2pi]`

Now let us suppose that `z` has the rectangular form:

`z = x+iy`

Equating real and imaginary components, we have:

`x=cosv`
`y = sinv`

So, the locus of the point `z` satisfies:

`x^2 + y^2 = cos^2v+sin^2v = 1`

Hence the point `z`, or `cosv + isinv` lies on the unit circle,QED