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

Oct 5, 2017

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

$z = \cos v + i \sin v \setminus \setminus$ where $v \in \left[0 , 2 \pi\right]$

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

$z = x + i y$

Equating real and imaginary components, we have:

$x = \cos v$
$y = \sin v$

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

${x}^{2} + {y}^{2} = {\cos}^{2} v + {\sin}^{2} v = 1$

Hence the point $z$, or $\cos v + i \sin v$ lies on the unit circle,QED