How to find the polar form of the complex number if given the rectangular form 7 - 5j?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The rectangular form of a complex number is:

z = x + i*y, where x represents the real part and y the imaginary part.

The polar form of a complex number is written in terms of a distance from origin of coordinate system and an angle measured from the positive x axis.

`z = r(cos theta + i*sin theta)`

The length of the vector r can be found using the real and imaginary parts of the complex number, which are given in rectangular form.

We'll use Pythagorean theorem to determine the length of the vector r, which is the hypotenuse of the right angle triangle, whose legs are the real and imaginary parts.

`r^2 = x^2 + y^2`

`r = sqrt(x^2 + y^2)`

In this case, we'll identify x and y:

x = 7 and y = -5.

`r = sqrt(7^2 + (-5)^2)`

`r = sqrt(74)`

`` Now, we'll find the angle `theta`  made by r to positive x axis.

`tan theta = y/x`

`tan theta = -5/7`

`theta = arctan (-5/7)`

`theta = 324.48 degrees`

Therefore, the polar form of the given complex number is:

`z = (sqrt(74))(cos 324.48 + isin324.48).`

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