Question

Write the complex number in polar form. express the argument in degrees, rounded to the nearest tenth, if necessary. 5) 2 +2i

Answer 1

Answer:

$z=\sqrt{8} (cos(45\°) + isin(45\°))$

Step-by-step explanation:

The polar form of a complex number is

$z=r(cos\theta + isin\theta)$

Where $r=\sqrt{a^{2} +b^{2} }$, $a=rcos\theta$, $b=rsin\theta$ and $\theta = tan^{-1}(\frac{b}{a} )$

In this case, we have the number $2+2i$, where $a=2$ and $b=2$, so the module $r$ is

$r=\sqrt{a^{2} +b^{2} }=\sqrt{2^{2} +2^{2} }=\sqrt{4+4}=\sqrt{8}$

And the angle is

$\theta = tan^{-1}(\frac{2}{2} )\\\theta=tan^{-1}(1)\\\theta = 45\°$

Therefore, the polar form of the given complex number is

$z=\sqrt{8} (cos(45\°) + isin(45\°))$

Answer 2

Here you need to calculate the magnitude and phase angle of the quantity 2 + 2i.

The magnitude is found using the Pyth. Thm. and is sqrt(2^2 + 2^2), or $\sqrt{8}$, or 2$\sqrt{2}$

Recognize that the phase angle is 45 deg, or $\pi$/4 rad.

Thus, the complex form is (2sqrt(2) angle $\pi$/4   rad ).