Problem 73 - 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. \([5(\cos\ 20^{\circ} + i\ \sin\ 20^{\circ})]^3\)

Step-by-Step Solution

Answer

The indicated power of the complex number in standard form is \(62.5 + 108.25i\).

Step 1: Analyze the Given Complex Number

We have a complex number in the polar form: \(5(\cos 20^{\circ} + i\sin 20^{\circ})^3\). Here, the magnitude (r) is 5, the initial angle (\(\Theta\)) is 20 degrees and the power (n) is 3.

Step 2: Use DeMoivre's Theorem

By DeMoivre's theorem, which is \((\cos x + i\sin x)^{n} = \cos(nx) + i\sin(nx)\), we can find \(5^3(\cos 20^{\circ} + i\sin 20^{\circ})^3\), which allow us to calculate the complex number to the power of 3.

Step 3: Simplify the Result

When we apply DeMoivre's theorem, we find \(5^3 (\cos 60^{\circ} + i\sin 60^{\circ}) = 125 (\cos 60^{\circ} + i\sin 60^{\circ})\). At this point, we can convert it back to the rectangular form. Therefore, \(125\cos 60^{\circ} = 125(\frac{\sqrt{3}}{2})\) and \(125\sin 60^{\circ} = 125 (\frac{1}{2})\), which yields \(62.5 + 108.25i\). So, the complex number in standard form is \(62.5 + 108.25i\).

Key Concepts

Polar Form

The polar form of a complex number is a way of representing complex numbers in terms of magnitude and angle. This method uses trigonometry to express a complex number. Instead of the usual Cartesian coordinates, we use the radius (or magnitude) and an angle from the positive x-axis.
In polar form, a complex number is represented as \[ r(\cos \theta + i \sin \theta) \]This equation tells us that:

• \(r\) is the magnitude or modulus of the complex number. It's the distance from the origin to the point on the complex plane.
• \(\theta\) is the angle the line makes with the positive real axis, commonly called the argument.
• The expression \(i\sin \theta\) signifies the imaginary component.

Using polar form can simplify many calculations involving complex numbers, especially those involving multiplications or powers, by transforming them into simpler trigonometric operations.

Complex Numbers

Complex numbers are numbers that comprise a real part and an imaginary part. They are commonly written in the form \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component, with \(i\) being the imaginary unit defined as \(i^2 = -1\).

Complex numbers allow us to solve equations that have no real solutions. For instance, the equation \(x^2 + 1 = 0\) has no real solutions since no real number squared gives -1. However, it does have complex solutions: \(x = i\) and \(x = -i\).
By representing numbers in the complex plane, complex numbers enable us to visualize arithmetic operations geometrically, and they play a crucial role in engineering and physics.

Standard Form Conversion

Converting from polar form back to standard rectangular form is an essential step in expressing complex numbers for practical applications. To convert a complex number from polar form \(r(\cos \theta + i \sin \theta)\) into the standard form \(a + bi\), we utilize the basic trigonometric identities:

• The real part: \(a = r \cos \theta\)
• The imaginary part: \(b = r \sin \theta\)

In the original exercise, we saw this conversion being used to get \(62.5 + 108.25i\) from the polar form \(125(\cos 60^{\circ} + i \sin 60^{\circ})\).

For practical calculations, remember that after finding the polar expression, use cosine to determine the real part and sine for the imaginary part. This conversion helps to present the final answer in a more broadly accepted format, being particularly helpful for graphing or further calculations.