Problem 56 - College Trigonometry

Question

Use DeMoivre's Theorem to find the indicated power of the given complex number. Express your final answers in rectangular form. $$ (\sqrt{3}+i)^{4} $$

Step-by-Step Solution

Answer

The result in rectangular form is 6 + 8\sqrt{3}i.

Step 1: Convert to polar form

Identify the modulus and argument of the complex number. The modulus is given by \(|z| = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = 2\). The argument \(\theta\) is \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\), which equals \(\frac{\pi}{6}\) radians, since \(\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}\). Therefore, the polar form is \(2 \text{cis} \frac{\pi}{6}\).

Step 2: Apply DeMoivre's Theorem

Use DeMoivre's Theorem which states \((r \text{cis} \theta)^n = r^n \text{cis}(n\theta)\). We have \(r = 2\) and \(n = 4\). Therefore, the expression becomes \((2 \text{cis} \frac{\pi}{6})^4 = 2^4 \text{cis}(4 \times \frac{\pi}{6}) = 16 \text{cis}(\frac{2\pi}{3})\).

Step 3: Convert back to rectangular form

The rectangular form can be found using \(z = r(\cos \theta + i \sin \theta)\). Thus, \(16 \text{cis}(\frac{2\pi}{3}) = 16(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3})\). We know that \(\cos \frac{2\pi}{3} = -\frac{1}{2}\) and \(\sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}\). Therefore, the rectangular form is \(-16 \times \frac{1}{2} + 16 \times \frac{\sqrt{3}}{2} i = -8 + 8\sqrt{3}i\).

Key Concepts

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to a two-dimensional complex plane. They are composed of a real part and an imaginary part. The general form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).

• **Real Part**: The \(a\) component in \(a + bi\) represents the real part.
• **Imaginary Part**: The \(b\) component is multiplied by \(i\) and forms the imaginary part of the complex number.
• **Imaginary Unit**: Denoted by \(i\), it satisfies the equation \(i^2 = -1\).

In the given problem, the complex number \(\sqrt{3} + i\) has a real part of \(\sqrt{3}\) and an imaginary part of \(1\).
Understanding complex numbers is crucial for performing operations such as addition, subtraction, and finding powers using techniques like DeMoivre's Theorem.

Polar Form

Complex numbers can also be expressed in polar form, which relates the complex number's modulus and argument to its rectangular representation. The polar form is often written as \(r \text{cis} \theta\), where:

• \(r\) is the modulus or magnitude of the complex number.
• \(\theta\) is the argument or angle in radians from the positive x-axis.
• The term \(\text{cis}\) is shorthand for \(\cos \theta + i \sin \theta\).

In polar form, the notation \(r \text{cis} \theta\) simplifies multiplication and powers of complex numbers.
In the exercise, \(\sqrt{3} + i\) is converted to polar form as follows:

• First, the modulus is calculated as \(2\).
• Then, the angle is determined as \(\frac{\pi}{6}\) using inverse tangent.

Thus, the complex number in polar form is \(2 \text{cis} \frac{\pi}{6}\).
Expressing complex numbers in polar form is particularly useful when applying DeMoivre's Theorem.

Rectangular Form

Rectangular form, also known as Cartesian form, is the standard way of writing complex numbers as \(a + bi\). This form directly translates to a position on the complex plane. While this form is intuitive for basic arithmetic operations, it is less convenient for multiplication or division compared to polar form.

• **Conversion**: To convert from polar to rectangular form, use the relationships \(a = r \cos \theta\) and \(b = r \sin \theta\).
• **Final Representation**: After applying DeMoivre's Theorem, complex numbers are often converted back to rectangular form for a more familiar representation.

In our solution, after using DeMoivre's Theorem, the complex number \(16 \text{cis}(\frac{2\pi}{3})\) is converted back to rectangular form. This results in the expression \(-8 + 8\sqrt{3}i\).
Mastering both forms and how to switch between them is essential for efficiently solving complex number problems.

Modulus and Argument

The modulus and argument are key components in understanding the polar form of a complex number.

• **Modulus (r)**: This is the distance from the origin to the point \((a, b)\) on the complex plane. It is calculated using the formula \(r = \sqrt{a^2 + b^2}\).
• **Argument (\(\theta\))**: The angle made with the positive x-axis, typically in radians, indicates the direction of the complex number from the origin. It can be calculated using \(\tan^{-1}(b/a)\).

The modulus provides the magnitude, while the argument gives the direction.
In the given problem, the modulus of \(\sqrt{3} + i\) was found as \(2\), and the argument was \(\frac{\pi}{6}\), making the polar form \(2 \text{cis} \frac{\pi}{6}\). Understanding these components is crucial when converting between polar and rectangular forms and in utilizing DeMoivre's Theorem effectively.