Problem 54 - College Trigonometry

Question

Use DeMoivre's Theorem to find the indicated power of the given complex number. Express your final answers in rectangular form. $$ (-\sqrt{3}-i)^{3} $$

Step-by-Step Solution

Answer

The result is -8i.

Step 1: Convert to Polar Form

To use DeMoivre's Theorem, convert the complex number \(-\sqrt{3}-i\) to polar form. This involves finding the modulus and the argument. The modulus, \(r\), is calculated as \(r = \sqrt{(-\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = 2\). The argument \(\theta\) is given by \(\tan^{-1}\left(\frac{-1}{-\sqrt{3}}\right)\), which simplifies to \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\). This angle corresponds to \(\frac{\pi}{6}\), but since the complex number is in the third quadrant, \(\theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6}\). So, the polar form is \(2\text{cis}\frac{7\pi}{6}\) where \(\text{cis}\theta = \cos\theta + i\sin\theta\).

Step 2: Apply DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form \(r \text{cis} \theta\), its nth power is given by \((r^n) \text{cis}(n\theta)\). For \((-\sqrt{3}-i)^3\), use \(r=2\), \(\theta=\frac{7\pi}{6}\), and \(n=3\). Compute \(r^n = 2^3 = 8\) and \(n\theta = 3 \times \frac{7\pi}{6} = \frac{21\pi}{6} = \frac{7\pi}{2}\). So, the result in polar form is \(8\text{cis}\frac{7\pi}{2}\).

Step 3: Simplify the Argument

The argument \(\frac{7\pi}{2}\) is outside the standard range \([0, 2\pi]\) for polar coordinates. Subtract \(2\pi\) multiples until the angle is within range: \(\frac{7\pi}{2} - 2\pi = \frac{7\pi}{2} - \frac{4\pi}{2} = \frac{3\pi}{2}\). Thus, the simplified polar form becomes \(8\text{cis}\frac{3\pi}{2}\).

Step 4: Convert Back to Rectangular Form

Convert \(8\text{cis}\frac{3\pi}{2}\) back to rectangular form. This involves using \(\cos\frac{3\pi}{2} = 0\) and \(\sin\frac{3\pi}{2} = -1\). Therefore, the rectangular form is \(8(0 + i(-1)) = 8(-i) = -8i\).

Key Concepts

Understanding Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. They are usually expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, with \(i\) representing the imaginary unit, which is defined as \(\sqrt{-1}\). Complex numbers are essential in various fields of engineering, physics, and mathematics.

They extend the idea of one-dimensional real numbers to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

Some key points about complex numbers include:

• The real and imaginary parts are both real numbers.
• The imaginary unit \(i\) satisfies \(i^2 = -1\).
• Complex numbers can be added, subtracted, multiplied, and divided using arithmetic rules, but always keeping in mind the property of \(i\).

Understanding these properties is foundational for using complex numbers in further calculations, as shown in DeMoivre's Theorem.

Exploring Polar Form

The polar form of a complex number is a way of representing it in terms of a modulus and an argument (angle). This form is particularly useful for multiplication, division, and exponentiation of complex numbers. Its format is \(r \text{cis} \theta\), where \(r\) is the modulus and \(\theta\) is the argument. The expression \( \text{cis} \theta\) refers to \(\cos \theta + i \sin \theta\).

Converting to Polar Form

To convert a complex number from rectangular form \((a + bi)\) to polar form:

• Calculate the modulus \(r = \sqrt{a^2 + b^2}\).
• Determine the argument \(\theta = \tan^{-1}(\frac{b}{a})\). Adjust the angle depending on the quadrant the number is in.

Using polar form simplifies the process of raising a complex number to a power, which is why it's used in conjunction with DeMoivre's Theorem.

Rectangular Form Revisited

Initially, complex numbers are often expressed in rectangular form, written as \(a + bi\). This format is straightforward, displaying both the real and imaginary components directly. While it is intuitive for addition and subtraction, it can be less convenient for multiplication, division, and taking powers compared to polar form.

Converting back to rectangular form from polar form involves using trigonometry:

• The real part is found using \( r \cos \theta \).
• The imaginary part is determined by \( r \sin \theta \).

For the example where the polar form is \(8\text{cis} \frac{3\pi}{2}\), we find:

• \(\cos \frac{3\pi}{2} = 0\), so the real component is 0.
• \(\sin \frac{3\pi}{2} = -1\), giving the imaginary component \(-8i\).

This results in the rectangular form \(-8i\). Understanding how to navigate between these forms is crucial for effectively using DeMoivre’s Theorem and similar calculations.

Modulus and Argument Explained

The modulus and argument are critical components in understanding and working with complex numbers in polar form.

The Modulus

The modulus of a complex number \(a + bi\) is essentially its "size" or "magnitude". It is calculated as \(r = \sqrt{a^2 + b^2}\). The modulus allows us to measure the distance of the number from the origin on the complex plane.

The Argument

The argument, on the other hand, is the angle \(\theta\) a complex number makes with the positive real axis. It is found using \(\tan \theta = \frac{b}{a}\), often requiring adjustments based on the quadrant. For example, if a complex number is in the third quadrant, as in the exercise, you would adjust \(\theta\) by adding \(\pi\) to the initial calculation.

These concepts are indispensable when expressing complex numbers in polar form and applying the beautiful DeMoivre's Theorem efficiently.