Argument of Complex Number/Examples/-3

From ProofWiki
Jump to navigation Jump to search

Example of Argument of Complex Number

$\arg \left({-3}\right) = \pi$


Proof

We have that:

$-3 = -3 + 0 i$

and so:

\(\displaystyle \left\lvert{-3}\right\rvert\) \(=\) \(\displaystyle \sqrt{3^2 + 0^2}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle 3\)


Hence:

\(\displaystyle \cos \left({\arg \left({-3}\right)}\right)\) \(=\) \(\displaystyle \dfrac {-3} 3\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle -1\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \arg \left({-3}\right)\) \(=\) \(\displaystyle \pi\) Cosine of Multiple of Pi


\(\displaystyle \sin \left({\arg \left({-3}\right)}\right)\) \(=\) \(\displaystyle \dfrac 0 3\) Definition of Argument of Complex Number
\(\displaystyle \) \(=\) \(\displaystyle 0\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \arg \left({-3}\right)\) \(=\) \(\displaystyle 0 \text { or } \pi\) Sine of Multiple of Pi


Hence:

$\arg \left({-3}\right) = \pi$

$\blacksquare$


Sources