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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 $z=a+bi$ 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 $\theta$ where $r$ is the length of the vector and $\theta$ is the angle made with the real axis.

From Pythagorean Theorem :

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

Past using the bones trigonometric ratios :

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

Multiplying each side past $r$ :

$r\cos \theta =a\text{and}r\sin \theta =b$

The rectangular form of a circuitous number is given past

$z=a+bi$ .

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

$\begin{array}{l}z=a+bi\\ =r\cos \theta +\left(r\sin \theta \right)i\\ =r\left(\cos \theta +i\sin \theta \right)\end{array}$

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

This can be summarized every bit follows:

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

Instance:

Limited the complex number in polar form.

$5+2i$

The polar class of a complex number $z=a+bi$ is $z=r\left(\cos \theta +i\sin \theta \right)$ .

And then, first find the absolute value of $r$ .

$\begin{array}{l}r=\left|z\right|=\sqrt{a^2+b^{\mathrm{two}}}\\ =\sqrt{{\mathrm{v}}^{\mathrm{ii}}+{\mathrm{two}}^2}\\ =\sqrt{25+4}\\ =\sqrt{29}\\ \approx 5.39\end{array}$

Now find the argument $\theta$ .

Since $a>0$ , use the formula $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$ .

$\begin{array}{l}\theta ={\tan }^{-1}\left(\frac{2}{5}\right)\\ \approx 0.38\end{array}$

Notation that hither $\theta$ is measured in radians.

Therefore, the polar form of $5+\mathrm{two}i$ is nearly $5.39\left(\cos \left(0.38\right)+i\sin \left(0.38\right)\right)$ .

Convert From Complex To Polar,

Source: https://www.varsitytutors.com/hotmath/hotmath_help/topics/polar-form-of-a-complex-number

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