Question 1208916
.
if z = cot (theta) - i , - pi < theta <0 , find | z |
~~~~~~~~~~~~~~~~~~~~~~


<pre>
|z|^2 = {{{cot^2(theta)}}} + {{{(-1)^2}}} = {{{cos^2(theta)/sin^2(theta)}}} + 1 = {{{(cos^2(theta)+sin^2(theta))/sin^2(theta)}}} = {{{1/sin^2(theta)}}}.


Hence,  |z|  is the square root of  {{{1/sin^2(theta)}}}.


Since  |z|  must be positive and  {{{sin(theta)}}}  is negative at  -{{{pi}}} < {{{theta}}} < 0,

we take the positive value of the square root of  {{{1/sin^2(theta)}}},  which is  {{{-1/sin(theta)}}} = {{{-csc(theta)}}}.


<U>ANSWER</U>.  At given conditions,  |z| = {{{-csc(theta)}}}.
</pre>

Solved.