Problem 57 - College Trigonometry

Question

Use DeMoivre's Theorem to find the indicated power of the given complex number. Express your final answers in rectangular form. $$ \left(\frac{5}{2}+\frac{5}{2} i\right)^{3} $$

Step-by-Step Solution

Answer

The result is \(-\frac{125}{4} + i\frac{125}{4}\).

Step 1: Convert to Polar Form

The complex number \( z = \frac{5}{2} + \frac{5}{2}i \) can be expressed in polar form. First, find the modulus \( r \): \( r = \sqrt{\left(\frac{5}{2}\right)^2 + \left(\frac{5}{2}\right)^2} = \frac{5}{\sqrt{2}} \). Next, find the argument \( \theta \): \( \tan^{-1}\left(\frac{\frac{5}{2}}{\frac{5}{2}}\right) = \tan^{-1}(1) = \frac{\pi}{4} \). So, \( z = \frac{5}{\sqrt{2}} \text{cis} \frac{\pi}{4} \).

Step 2: Apply DeMoivre's Theorem

DeMoivre's Theorem states that if \( z = r \text{cis} \theta \), then \( z^n = r^n \text{cis}(n\theta) \). Here, \( n = 3 \). So, \( z^3 = \left(\frac{5}{\sqrt{2}}\right)^3 \text{cis} \left(3 \times \frac{\pi}{4}\right) \).

Step 3: Simplify the Modulus

Calculate \( \left(\frac{5}{\sqrt{2}}\right)^3 = \frac{125}{2\sqrt{2}} = \frac{125\sqrt{2}}{4} \).

Step 4: Compute the Argument

The argument becomes \( 3 \times \frac{\pi}{4} = \frac{3\pi}{4} \). Thus, \( z^3 = \frac{125\sqrt{2}}{4} \text{cis} \frac{3\pi}{4} \).

Step 5: Convert Back to Rectangular Form

To convert to rectangular form, use the identities \( \cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \) and \( \sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2} \). Thus, \( z^3 = \frac{125\sqrt{2}}{4} \left(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = -\frac{125}{4} + i\frac{125}{4} \).

Key Concepts

Polar Form of Complex Numbers

In mathematics, complex numbers can be expressed in a form known as the polar form, which highlights their magnitude and direction on the complex plane. Every complex number has a "modulus," which is the distance from the origin of the complex plane to the point, and an "argument," which is the angle from the positive real axis to the line representing the complex number. This representation is especially handy for multiplication and exponentiation of complex numbers.

To find the polar form, you start by calculating the modulus, denoted as \( r \). For a complex number \( a + bi \), the modulus is calculated using the formula:

• \( r = \sqrt{a^2 + b^2} \)

Next, determine the argument \( \theta \) by calculating the angle using the inverse tangent function:

• \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \)

For example, the complex number \( \frac{5}{2} + \frac{5}{2}i \) has a modulus of \( \frac{5}{\sqrt{2}} \) and an argument of \( \frac{\pi}{4} \). Its polar form is thus \( \frac{5}{\sqrt{2}} \text{cis} \frac{\pi}{4} \), where "cis" stands for "cosine plus i sine."

This polar representation allows us to use DeMoivre's Theorem efficiently, as it becomes simple to raise complex numbers to powers or extract roots.

Rectangular Form of Complex Numbers

The rectangular form of complex numbers is the standard way of expressing them, especially helpful when dealing with addition and subtraction. In this form, a complex number is expressed as \( a + bi \), where \( a \) represents the real part and \( b \) represents the imaginary part. The complex number is shown as a point \( (a, b) \) in the complex plane.

In practical terms, understanding this form helps you visualize complex numbers as combinations of real and imaginary components.

To convert a complex number in polar form back to rectangular form, you utilize the fact that any polar coordinate \( r \text{cis} \theta \) can be expressed in terms of cosine and sine:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

Where in the provided exercise, the calculated results demonstrate once you utilized polar to rectangular conversion for \( z^3 = \frac{125\sqrt{2}}{4} \text{cis} \frac{3\pi}{4} \), resulting in \( -\frac{125}{4} + i\frac{125}{4} \). This rectangular form is often more intuitive for students to understand and visualize in graph format.

Converting Between Forms of Complex Numbers

Understanding how to convert complex numbers between polar and rectangular forms is essential, especially in the fields of engineering and physics, for simplifying complex arithmetic operations.

When converting from rectangular form to polar form:

• Find the modulus \( r \) using \( r = \sqrt{a^2 + b^2} \).
• Determine the argument \( \theta \) using \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

Then represent the complex number as \( r \text{cis} \theta \).

Conversely, when converting from polar form to rectangular form:

• Calculate the real part as \( a = r \cos \theta \).
• Calculate the imaginary part as \( b = r \sin \theta \).

The seamless conversion demonstrated in the exercise showcases the versatility of DeMoivre’s Theorem and the ease of performing exponentiations in polar form, later transformed back into rectangular to give easily interpretable results. Such exercises enhance comprehension of both forms' practicality and the vital role of geometry in complex number operations.