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

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Answer 1 · 2017-10-05T21:35:19

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

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