Problem 71 - Precalculus with Limits

Question

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ \left[5\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)\right]^{3} $$

Step-by-Step Solution

Answer

The result in standard form is \(62.5 + 108.25i\).

Step 1: Apply DeMoivre's Theorem

According to DeMoivre's Theorem, we can solve \(\left[5\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)\right]^{3}\) by raising the magnitude 5 to the power 3 and multiplying the angle 20 degrees by 3. This gives us \(125\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\).

Step 2: Convert to standard form

The standard form of a complex number is \(a + bi\). We can get this by multiplying the magnitude (125) with cos(60 degrees) and sin(60 degrees) respectively. Cos(60 degrees) equals \(0.5\) and sin(60 degrees) equals \(\sqrt{3}/2\). Thus, we get \(125 * 0.5 + 125 * \sqrt{3}/2 * i = 62.5 + 108.25i\).

Key Concepts

Understanding Complex Numbers

At the heart of understanding DeMoivre's Theorem lies a solid grasp of complex numbers. A complex number is a number that can be expressed in the form of a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i^2 = -1. Unlike real numbers, complex numbers include a real part and an imaginary part, allowing them to encapsulate rotation and magnitude in a two-dimensional space known as the complex plane.

The beauty of complex numbers is that they enable calculations involving roots of negative numbers, which are not possible within the set of real numbers alone, thereby expanding our realm of mathematics to new and intriguing operations and functions. When it comes to raising complex numbers to powers or extracting roots, DeMoivre's Theorem is a powerful tool that simplifies these processes by working with the polar form of complex numbers.

The Standard Form of a Complex Number

Diving deeper, let’s discuss the standard form of a complex number. It is represented as a + bi, where a is the real part and b is the imaginary part. The standard form is essentially an ordered pair on the complex plane, giving us a convenient way to express and calculate with complex numbers.

In many mathematical problems, including those involving DeMoivre's Theorem, you’ll often need to convert complex numbers from their polar or trigonometric form to this standard form to make them easier to understand and use in further calculations. Take our exercise for example: after using DeMoivre's Theorem, we converted the trigonometric representation 125(cos 60° + i sin 60°) into 62.5 + 108.25i by using the cosine and sine values to calculate the real and imaginary components respectively.

Magnitude and Angle of Complex Numbers

The magnitude (or modulus) and angle (or argument) of complex numbers are concepts originating from their representation in polar form. The magnitude of a complex number z = a + bi is its distance from the origin to the point (a, b) on the complex plane, calculated as |z| = √(a² + b²). The angle, on the other hand, is the counterclockwise angle a line from the origin to (a, b) makes with the positive real axis.

Why are these concepts important? When you use DeMoivre's Theorem to raise complex numbers to powers, you're working with their magnitudes and angles. In the exercise, raising the complex number to the third power meant cubing the magnitude (from 5 to 125) and multiplying the angle by three (from 20 degrees to 60 degrees), giving us a new magnitude and angle that describe the resulting complex number in polar form. To finalize our answer, we then convert this back into the standard form by using the calculated magnitude and angle to find the new real and imaginary components.