Convert From Complex To Polar

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Polar Form of a Complex Number

The polar form of a circuitous number is another manner to correspond a complex number. The form $z = a + b i$ is called the rectangular coordinate class of a complex number.

The horizontal centrality is the existent centrality and the vertical axis is the imaginary axis. Nosotros detect the real and complex components in terms of $r$ and $θ$ where $r$ is the length of the vector and $θ$ is the angle made with the real axis.

From Pythagorean Theorem :

$r^{2} = a^{2} + b^{2}$

Past using the bones trigonometric ratios :

$\cos θ = \frac{a}{r}$ and $\sin θ = \frac{b}{r}$ .

Multiplying each side past $r$ :

$r \cos θ = a and r \sin θ = b$

The rectangular form of a circuitous number is given past

$z = a + b i$ .

Substitute the values of $a$ and $b$ .

$\begin{array}{l} z = a + b i \\ = r \cos θ + (r \sin θ) i \\ = r (\cos θ + i \sin θ) \end{array}$

In the case of a complex number, $r$ represents the absolute value or modulus and the angle $θ$ is chosen the argument of the complex number.

This can be summarized every bit follows:

The polar form of a complex number $z = a + b i$ is $z = r (\cos θ + i \sin θ)$ , where $r = | z | = \sqrt{a^{two} + b^{2}}$ , $a = r \cos θ and b = r \sin θ$ , and $θ = \tan^{- 1} (\frac{b}{a})$ for $a > 0$ and $θ = \tan^{- 1} (\frac{b}{a}) + π$ or $θ = \tan^{- 1} (\frac{b}{a}) + 180 °$ for $a < 0$ .