SOLUTION: if z = cot (theta) - i , - pi < theta <0 , find | z |

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: if z = cot (theta) - i , - pi < theta <0 , find | z |      Log On


   

Question 1208916: if z = cot (theta) - i , - pi < theta <0 , find | z |
Answer by ikleyn(52909) About Me  (Show Source):
You can put this solution on YOUR website!
.
if z = cot (theta) - i , - pi < theta <0 , find | z |
~~~~~~~~~~~~~~~~~~~~~~ 

|z|^2 = cot%5E2%28theta%29 + %28-1%29%5E2 = cos%5E2%28theta%29%2Fsin%5E2%28theta%29 + 1 = %28cos%5E2%28theta%29%2Bsin%5E2%28theta%29%29%2Fsin%5E2%28theta%29 = 1%2Fsin%5E2%28theta%29.


Hence,  |z|  is the square root of  1%2Fsin%5E2%28theta%29.


Since  |z|  must be positive and  sin%28theta%29  is negative at  -pi < theta < 0,

we take the positive value of the square root of  1%2Fsin%5E2%28theta%29,  which is  -1%2Fsin%28theta%29 = -csc%28theta%29.


ANSWER.  At given conditions,  |z| = -csc%28theta%29.

Solved.