Polar Form of Complex Number/Examples/-i

From ProofWiki
Jump to navigation Jump to search

Example of Polar Form of Complex Number

The imaginary number $-i$ can be expressed in polar form as $\polar {1, \dfrac {3 \pi} 2}$.


Proof

\(\displaystyle \cmod {-i}\) \(=\) \(\displaystyle \sqrt {0^2 + \paren {-1}^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle 1\)


Then:

\(\displaystyle \map \cos {\map \arg {-i} }\) \(=\) \(\displaystyle \dfrac 0 1\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \arg {-i}\) \(=\) \(\displaystyle \dfrac \pi 2 \text { or } \dfrac {3 \pi} 2\) Cosine of Half-Integer Multiple of Pi


\(\displaystyle \map \sin {\map \arg {-i} }\) \(=\) \(\displaystyle \dfrac {-1} 1\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle -1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \arg {-i}\) \(=\) \(\displaystyle \dfrac {3 \pi} 2\) Sine of Half-Integer Multiple of Pi


Hence:

$\map \arg {-i} = \dfrac {3 \pi} 2$

and hence the result.

$\blacksquare$


Sources