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Chapter 6: Problem 18

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$-2+2 i \sqrt{3}$$

Short Answer

Expert verified
The polar form of the complex number -2 + 2i\( \sqrt{3} \) is 4(cos(2蟺/3) + i sin(2蟺/3)) or equivalently 4e^(2蟺i/3).

Step by step solution

01

Plotting the complex number

The complex number is -2 + 2i\( \sqrt{3} \). Hence, the x-coordinate (real part) is -2 and the y-coordinate (imaginary part) is 2\( \sqrt{3} \). This point can be plotted on the complex plane.
02

Finding the magnitude of the complex number

The magnitude (r) of a complex number is given by the formula \( \sqrt{x^2 + y^2} \). Substituting x=-2 and y=2\( \sqrt{3} \) into the formula, we get r = \( \sqrt{(-2)^2 + (2\sqrt{3})^2} \) = 4.
03

Finding the argument of the complex number

The argument (胃) is given by the formula \( \arctan{(y/x)} \). Substituting y=2\( \sqrt{3} \) and x=-2 into the formula, we get 胃 = \( \arctan{(-\sqrt{3})} \). Since the complex number is in the second quadrant, the argument would be 蟺 - \( \arctan{(\sqrt{3})} \) = 2蟺/3.
04

Writing the complex number in polar form

A complex number in polar form is usually written as r(cos 胃 + i sin 胃), or using Euler's formula, re^(i胃). Substituting r=4 and 胃=2蟺/3, we get the polar form of the complex number as 4(cos(2蟺/3) + i sin(2蟺/3)) or 4e^(2蟺i/3).

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