Problem 78 - 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. \((\cos\ 0 + i\ \sin\ 0)^{20}\)

Step-by-Step Solution

Answer

The standard form of the complex number \((\cos\ 0 + i\ \sin\ 0)^{20}\) is 1.

Step 1: Use DeMoivre's Theorem

According to DeMoivre's theorem, \((\cos\ \theta + i\ \sin\ \theta)^n = \cos (n \theta)+ i\ \sin (n \theta)\). Applying DeMoivre's theorem to the complex number \((\cos\ 0 + i\ \sin\ 0)^{20}\), multiply the angle 0 by the power 20, i.e., \(n \theta = 20 \times 0 = 0\).

Step 2: Find the results for cosine and sine

The value of \(\cos 0\) and \(\sin 0\) are 1 and 0 respectively. Therefore, applying these values, the solution becomes \(\cos 0 + i \sin 0\), which equates to \(1 + 0i\).

Step 3: Result in Standard form

Expressing the solution in standard form results in \(1 + 0i\), which simplifies to 1 (the imaginary part disappears as it is zero).

Key Concepts

Complex Numbers

Complex numbers are essential in mathematics and have the general form \(a + bi\), where \(a\) and \(b\) are real numbers. The term \(a\) is called the real part, and \(bi\) is the imaginary part. In this context, \(i\) is the imaginary unit, defined by the property that \(i^2 = -1\). Complex numbers are used because they allow for solutions to equations that do not have real solutions.

In the exercise given, the complex number is represented in polar form as \((\cos\ 0 + i\ \sin\ 0)\). Converting between this polar form and the standard form \(a + bi\) is an essential skill when dealing with complex numbers. The polar form is particularly useful for multiplication and taking powers, which is why we use it in combination with DeMoivre's Theorem.

Standard Form

In solving problems involving complex numbers, we often need to express our final answers in standard form. This form is \(a + bi\), where both \(a\) and \(b\) are real numbers.

From the exercise, after applying DeMoivre's Theorem, the expression \((\cos\ 0 + i \sin\ 0)^{20}\) results in \(\cos\ 0 + i \sin\ 0\). Given \(\cos 0 = 1\) and \(\sin 0 = 0\), the standard form becomes \(1 + 0i\), which simplifies to 1.

• \(a = 1\)
• \(b = 0\)

This reflects that the imaginary part is zero, leaving only the real part.

Powers of Complex Numbers

Raising complex numbers to a power can be simplified using DeMoivre's Theorem. This theorem simplifies complex calculations, especially when the complex number is in its polar form \((\cos\ \theta + i\ \sin\ \theta)\). The theorem states:

• If you're raising to a power \(n\), multiply the angle \(\theta\) by \(n\) and compute \(\cos(n\theta) + i\sin(n\theta)\).

For the exercise \((\cos 0 + i \sin 0)^{20}\), applying DeMoivre’s Theorem results in:

• \(n \theta = 20 \times 0 = 0\)
• \(\cos(0) + i\sin(0)\ = 1 + 0i\)

This demonstrates how DeMoivre's Theorem provides a straightforward path to finding the power of a complex number, eventually giving us a simple real number answer.