# Complex numbers: express in modulus-argument form (cis)

• Feb 15th 2011, 08:51 PM
kumquat
Complex numbers: express in modulus-argument form (cis)
1 + cot(x)i , express in modulus argument form where 0 < x < pi/2

I understand how to get the modulus by finding (1 + cot^2(x))^1/2 = cosec(x)

but the answer gives cosec(x)cis(pi/2 - x), i dont understand why the (pi/2 - x) is needed and not just cis(x).
• Feb 15th 2011, 09:01 PM
Prove It
Length: $\displaystyle \sqrt{1 + \cot^2{x}} = \sqrt{\csc^2{x}} = \csc{x}$.

Angle: $\displaystyle \arctan{\left(\frac{\cot{x}}{1}\right)} = \arctan{(\cot{x})} = \arctan{\left[\tan{\left(\frac{\pi}{2} - x\right)}\right]} = \frac{\pi}{2} - x$.