Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-12: Use De Moivre's theorem to change the given complex number to the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-1+i)^{8} $$

Step-by-Step Solution

Answer

The result is 16.

Step 1: Express the Complex Number in Polar Form

The given complex number is \(-1 + i\). First, express this number in polar form. The modulus \(r\) of the complex number is given by \(r = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\). The argument \(\theta\) is \(\tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1) = \frac{3\pi}{4}\) radians, since the number lies in the second quadrant. Thus, the polar form of \(-1 + i\) is \(\sqrt{2}(\cos \frac{3\pi}{4} + i\sin \frac{3\pi}{4})\).

Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem, \((r(\cos \theta + i\sin \theta))^n = r^n (\cos(n\theta) + i\sin(n\theta))\). We have \(r = \sqrt{2}\), \(\theta = \frac{3\pi}{4}\), and \(n = 8\). So, we need to compute \(r^8 = (\sqrt{2})^8 = 16\). Also compute \(n\theta = 8 \times \frac{3\pi}{4} = 6\pi\). Therefore, \((-1+i)^8 = 16(\cos(6\pi) + i\sin(6\pi))\).

Step 3: Simplify the Trigonometric Expressions

We know that \(\cos(6\pi) = 1\) and \(\sin(6\pi) = 0\) because \(6\pi\) corresponds to a full circle. Substitute these into the expression: \((-1+i)^8 = 16(1 + i\cdot0) = 16 + 0i\).

Step 4: Express in Standard Form

The expression \(16 + 0i\) is already in the standard form \(a + bi\) with \(a = 16\) and \(b = 0\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by \(i^2 = -1\).
These numbers extend the familiar concept of real numbers and they are plotted on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Using this plane, each complex number corresponds to a unique point or position in a 2D plane.
Some key points about complex numbers include:

• The **real part** \(a\) is the component without \(i\).
• The **imaginary part** \(b\) refers to the coefficient of \(i\).
• Complex numbers can also be expressed in **polar form**, which makes multiplying and powering them easier.

Understanding complex numbers is essential when applying De Moivre's Theorem, which allows us to compute powers and roots of complex numbers more effectively.

Polar Form

Polar form is an alternative representation of complex numbers, which is particularly useful for multiplication and powering. In this form, a complex number is expressed as \(r(\cos \theta + i\sin \theta)\).
Here, \(r\) is the modulus and \(\theta\) is the argument:

• The **modulus** \(r\) is the distance of the complex number from the origin on the complex plane, calculated by \(r = \sqrt{a^2 + b^2}\).
• The **argument** \(\theta\) is the angle made with the positive real axis, often found using the inverse tangent function.

Converting a complex number to polar form involves computing these two components. For example, the modulus of \(-1 + i\) is \(\sqrt{2}\), and the argument is \(\frac{3\pi}{4}\) radians, since it lies in the second quadrant.
This representation simplifies operations like multiplication and exponentiation, which would otherwise be cumbersome in standard form.

Trigonometric Functions

Trigonometric functions, specifically sine and cosine, are crucial when working with complex numbers in polar form. These functions help to govern the rotation and describe the position of a point on the unit circle.
In the context of complex numbers and De Moivre's Theorem, they represent how the angle (argument) is spread out over the unit circle. For any angle \(\theta\):

• **Cosine** function, \(\cos \theta\), indicates the horizontal position on the unit circle.
• **Sine** function, \(\sin \theta\), indicates the vertical position on the unit circle.

Using De Moivre’s Theorem, \((r(\cos \theta + i \sin \theta))^n\) simplifies to \(r^n(\cos(n\theta) + i \sin(n\theta))\), allowing easy computation of powers of complex numbers. For example, multiplying the angle by \(n\) effectively applies the rotation \(n\) times, and geometric properties of the sine and cosine lead to simplifications like \(\cos(6\pi) = 1\) and \(\sin(6\pi) = 0\). This eventually yields a simple result in real/imaginary terms, such as \(16 + 0i\) in our exercise.