Problem 32 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ -3-3 \sqrt{3} i $$

Step-by-Step Solution

Answer

Polar form: \[6\left(\cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right)\right)\].

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(-3 - 3 \sqrt{3} i \).Here, the real part \(a\) is \(-3\) and the imaginary part \(b\) is \(-3 \sqrt{3}\).

Step 2: Calculate the Magnitude

The magnitude \(r\) of the complex number is calculated using the formula:\[r = \sqrt{a^2 + b^2}\]Substitute \(a = -3\) and \(b = -3 \sqrt{3}\) into the equation:\[r = \sqrt{(-3)^2 + (-3 \sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6\]

Step 3: Find the Argument \(\theta\)

The argument \(\theta\) is given by the formula:\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]Substituting the values \(a = -3\) and \(b = -3 \sqrt{3}\):\[\theta = \tan^{-1}\left(\frac{-3 \sqrt{3}}{-3}\right) = \tan^{-1}(\sqrt{3})\]The angle corresponding to \(\tan^{-1}(\sqrt{3})\) is \(\frac{\pi}{3}\).However, since the complex number is in the third quadrant (both parts are negative), the actual angle is:\[\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}\]

Step 4: Write the Polar Form

The polar form of a complex number is written as:\[r \left( \cos(\theta) + i\sin(\theta) \right)\]Substituting the values \(r = 6\) and \(\theta = \frac{4\pi}{3}\):\[6 \left( \cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right) \right)\]

Key Concepts

Polar Form

Let's dive into the polar form of complex numbers. Polar form is a way of expressing complex numbers in terms of their magnitude and angle, also known as the argument. Instead of a standard form like \(a + bi\), where \(a\) and \(b\) are real numbers, polar form is represented as \(r(\cos(\theta) + i\sin(\theta))\). This notation emphasizes the geometric interpretation of complex numbers as vectors in a plane, offering insights into both their size and direction.

• \(r\) stands for the magnitude, a measure of the vector's length.
• \(\theta\) represents the argument, indicating the angle the vector forms with the positive x-axis.

Converting to polar form helps in simplifying multiplication and division operations between complex numbers, as it utilizes the properties of trigonometric identities effectively.

Magnitude

The magnitude of a complex number highlights its size or length when depicted as a vector in the complex plane. For a complex number \(a + bi\), the magnitude \(r\) is calculated using the formula:\[r = \sqrt{a^2 + b^2}\]This is derived from the Pythagorean theorem, viewing the real part \(a\) and the imaginary part \(b\) as forming a right triangle on a Cartesian plane.

In our example where the complex number is \(-3 - 3\sqrt{3}i\), plugging into the formula gives:\[r = \sqrt{(-3)^2 + (-3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6\]This 6 units measurement indicates how far the complex number is from the origin in the complex plane.

Argument

The argument of a complex number is the angle it forms with the positive x-axis in the complex plane. It is denoted by \(\theta\) and can be calculated using the formula for tangent:\[\theta = \tan^{-1}\left( \frac{b}{a} \right)\]For our complex number \(-3 - 3\sqrt{3}i\), substituting the values yields:\[\theta = \tan^{-1}\left(\frac{-3\sqrt{3}}{-3}\right) = \tan^{-1}(\sqrt{3})\]This corresponds to \(\frac{\pi}{3}\), but since the complex number lies in the third quadrant, it’s essential to add \(\pi\) to the angle to reflect this quadrant shift:\[\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}\]

The argument is crucial in understanding the direction of the complex number on the plane.

Third Quadrant

The third quadrant in the complex plane plays a key role when determining both the direction and sign of a complex number's components. Quadrants are used to denote the area where a complex number lies in relation to the real and imaginary axes.

In the third quadrant:

• Both real and imaginary parts of the complex number are negative.
• The angle (argument) should be calculated with adjustments since standard angle results will not automatically reflect the negative directions in both axes.

For the complex number \(-3 - 3\sqrt{3}i\), both components are negative, confirming its position in the third quadrant. Calculations must consider this placement to ensure the argument accurately reflects this orientation, hence the adjustment of adding \(\pi\) to the calculated angle to ensure it lies correctly on the complex plane.