Polar Form of Complex Number/Examples/-5 + 5i

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Example of Polar Form of Complex Number

The complex number $-5 + 5 i$ can be expressed as a complex number in polar form as $\polar {5 \sqrt 2, \dfrac {3 \pi} 4}$.


Proof

-5 + 5 i.png
\(\displaystyle \cmod {-5 + 5 i}\) \(=\) \(\displaystyle \sqrt {\paren {-5}^2 + 5^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {2 \times 25}\)
\(\displaystyle \) \(=\) \(\displaystyle 5 \sqrt 2\)


Then:

\(\displaystyle \map \cos {\map \arg {-5 + 5 i} }\) \(=\) \(\displaystyle \dfrac {-5} {5 \sqrt 2}\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle -\frac {\sqrt 2} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \arg {-5 + 5 i}\) \(=\) \(\displaystyle \pm \dfrac {3 \pi} 4\) Cosine of $135 \degrees$, Cosine of $225 \degrees$


\(\displaystyle \map \sin {\map \arg {-5 + 5 i} }\) \(=\) \(\displaystyle \dfrac 5 {5 \sqrt 2}\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle \frac {\sqrt 2} 2\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \arg {-5 + 5 i}\) \(=\) \(\displaystyle \dfrac \pi 4 \text { or } \dfrac {3 \pi} 4\) Sine of $45 \degrees$, Sine of $135 \degrees$


Hence:

$\map \arg {-5 + 5 i} = \dfrac {3 \pi} 4$

and hence the result.

$\blacksquare$


Sources