Problem 21 - Algebra and Trigonometry

Question

In Exercises \(11-26,\) plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$ -3 \sqrt{2}-3 i \sqrt{3} $$

Step-by-Step Solution

Answer

Polar form of the given complex number in radians: \(6[cos(\frac{4\pi}{3}) + isin(\frac{4\pi}{3})]\), and in degrees: \(6[cos(240^{\circ}) + isin(240^{\circ})]\).

Step 1: Identifying the real and imaginary parts

We can identify that the real part of the given complex number is -3\(\sqrt{2}\) and the imaginary part is -3\(\sqrt{3}\).

Step 2: Calculating the magnitude (r)

The magnitude r of the complex number (distance from the origin to the point in the complex plane) is given by \(r=\sqrt{x^2+y^2}\). After replacing x with -3\(\sqrt{2}\) and y with -3\(\sqrt{3}\), we get \(r=\sqrt{(-3\sqrt{2})^2 + (-3\sqrt{3})^2}\), so \(r=6\).

Step 3: Calculating the argument (\(\Theta\))

The argument \(\Theta\), or the angle that the ray from the origin to the point makes with the positive x-axis, is given by \(\Theta = atan2(y, x)\). Here, atan2 is the two-argument variant of the arctangent function. By substituting y = -3\(\sqrt{3}\) and x = -3\(\sqrt{2}\) in \(\Theta = atan2(y, x)\), we get \(\Theta=\frac{4\pi}{3}\) (in radian), or \(240^{\circ}\) (in degree).

Step 4: Writing the complex number in polar form

The polar form of a complex number is represented as \( r(cos(\Theta) + isin(\Theta))\). By substituting r = 6 and \(\Theta = \frac{4\pi}{3}\) (or \(240^{\circ}\) in degree), we get the polar form of the complex number as \(6[cos(\frac{4\pi}{3}) + isin(\frac{4\pi}{3})]\) in radians or \(6[cos(240^{\circ}) + isin(240^{\circ})]\) in degrees.

Key Concepts

Plotting Complex Numbers

When we talk about plotting complex numbers, we are referring to the process of representing a complex number on a two-dimensional plane known as the complex plane. This plane has two axes: the horizontal axis (real axis) and the vertical axis (imaginary axis). Each complex number corresponds to a unique point on this plane, where the horizontal coordinate is the real part of the number and the vertical coordinate is the imaginary part.

To plot the complex number \( -3 \sqrt{2} - 3i \sqrt{3} \), we find the point that is \( -3 \sqrt{2} \) units along the real axis and \( -3 \sqrt{3} \) units along the imaginary axis. Since both values are negative, the point lies in the third quadrant of the complex plane. Visualizing this plot can help students understand the position and relation of complex numbers to each other.

Magnitude of Complex Numbers

The magnitude of a complex number is a measure of its distance from the origin of the complex plane. To calculate the magnitude, which is also known as the absolute value or the modulus of a complex number, we use the formula \(r = \sqrt{x^2 + y^2}\), where \(x\) is the real part and \(y\) is the imaginary part of the complex number.

For our example \( -3 \sqrt{2} - 3i \sqrt{3} \), the magnitude is computed as \(r = \sqrt{(-3\sqrt{2})^2 + (-3\sqrt{3})^2}\), leading to \(r = 6\). The magnitude gives us an idea of the size or length of the complex number vector in the complex plane.

Argument of Complex Numbers

The argument of a complex number is the angle made with the positive real axis by the line connecting the origin to the point representing the complex number in the complex plane. This angle is measured counterclockwise and can be found using the arctangent function.

In our given problem, the argument \(\Theta\) of the complex number \( -3 \sqrt{2} - 3i \sqrt{3} \) is calculated using the function of two arguments, \(atan2(y, x)\). By inserting \(x = -3 \sqrt{2}\) and \(y = -3 \sqrt{3}\), we obtain \(\Theta = \frac{4\pi}{3}\) radians or \(240^{\textrm{°}}\) degrees. This angle is essential when expressing the complex number in polar form and helps us to understand the orientation of the complex number in the complex plane.

Polar Coordinates

Polar coordinates provide an alternative way of describing the position of a point in a plane, using the distance from the origin (magnitude) and the angle from the positive real axis (argument). A complex number in polar form is expressed as \( r(\cos(\Theta) + i\sin(\Theta)) \).

This representation is particularly useful when performing multiplications or divisions of complex numbers, as it simplifies the process. In the case of our complex number \( -3 \sqrt{2} - 3i \sqrt{3} \), the polar form is \(6[\cos(\frac{4\pi}{3}) + i\sin(\frac{4\pi}{3})]\) in radians, or \(6[\cos(240^{\textrm{°}}) + i\sin(240^{\textrm{°}})]\) in degrees. By understanding the polar coordinates of complex numbers, students can gain a deeper insight into the geometric properties of complex numbers and how they relate to one another.