Problem 107 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(1+i)^{3}$$

Step-by-Step Solution

Answer

-2 + 2i

Step 1: Convert to Polar Form

Convert the complex number \(1+i\) into polar form. The magnitude \(r\) of the complex number is given by \(\sqrt{1^2 + 1^2}= \sqrt{2}\). The angle \(\theta\) is given by \( \arctan \left(\frac{1}{1}\right) = \frac{\pi}{4}\). So, in polar form, the complex number is written as \(\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}))\).

Step 2: Apply De Moivre's Theorem

Apply De Moivre's theorem to find the cube of the complex number. This yields \((\sqrt{2}^3)(\cos(3*\frac{\pi}{4}) + i\sin(3*\frac{\pi}{4})) = 2\sqrt{2}(\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}))\).

Step 3: Convert Back to Rectangular Form

Now we convert the result from polar form back to rectangular form. The cosine of \(\frac{3\pi}{4}\) equals \(-\frac{\sqrt{2}}{2}\) and the sine of \(\frac{3\pi}{4}\) equals \(\frac{\sqrt{2}}{2}\). So, the result in rectangular form is \(2\sqrt{2}(-\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}) = -2 + 2i\).

Key Concepts

Polar Form

Polar form is a way of expressing complex numbers using a magnitude and an angle. Here’s how it works:
Imagine your complex number as a point on a plane, where the x-axis is the real component and the y-axis is the imaginary component.

• The magnitude, often denoted as \(r\), tells us how far the point is from the origin.
• The angle, represented as \(\theta\), indicates the direction of the point from the positive x-axis.

To convert a complex number \(a + bi\) into polar form, you:

• Calculate the magnitude using \(r = \sqrt{a^2 + b^2}\).
• Find the angle with \(\theta = \arctan(b/a)\).

This results in the polar representation as \(r(\cos(\theta) + i\sin(\theta))\). For example, for the complex number \(1+i\), the magnitude \(r\) is \(\sqrt{2}\) and \(\theta\) is \(\frac{\pi}{4}\), giving \(\sqrt{2}(\cos(\frac{\pi}{4}) + i\sin(\frac{\pi}{4}))\). Polar form is particularly useful for multiplying and dividing complex numbers, as well as finding powers and roots using De Moivre's Theorem.

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part, usually written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).
These numbers exist because there are equations like \(x^2 + 1 = 0\) that cannot be solved within the realm of real numbers.When working with complex numbers:

• The real part, \(a\), lies on the horizontal axis.
• The imaginary part, \(b\), lies on the vertical axis.

Complex numbers can be visualized on the complex plane, which makes them easier to understand and manipulate using geometric techniques. Additionally, they are crucial in various fields such as engineering, physics, and signal processing. Using algebraic operations on complex numbers involves:

• Addition and subtraction by combining like terms (real with real, imaginary with imaginary).
• Multiplication using distributive property, where \(i^2\) is replaced with \(-1\).

De Moivre's Theorem further extends their usefulness by allowing for the computation of powers and roots of complex numbers with ease when expressed in polar form.

Rectangular Form

Rectangular form is the standard way of expressing complex numbers, written as \(a + bi\). This form is straightforward and is typically used for arithmetic operations involving complex numbers.
In order to effectively convert from polar to rectangular form:- You need to know the polar representation \(r(\cos(\theta) + i\sin(\theta))\).- Use the formulas \(a = r\cos(\theta)\) and \(b = r\sin(\theta)\) to find your rectangular coordinates.For instance, if a complex number is given in polar form as \(2\sqrt{2} (\cos(\frac{3\pi}{4}) + i\sin(\frac{3\pi}{4}))\), we use:

• \(a = 2\sqrt{2} \times \cos(\frac{3\pi}{4}) = -2\)
• \(b = 2\sqrt{2} \times \sin(\frac{3\pi}{4}) = 2\)

Thus, the rectangular form becomes \(-2 + 2i\). Understanding both forms—polar and rectangular—is useful, as each has its strengths in different mathematical operations. Rectangular form is more intuitive for basic arithmetic, while polar form simplifies multiplication and division.