Problem 73 - Algebra and Trigonometry

Question

\(65-76=\) Find the indicated power using DeMoivre's Theorem. $$ (-1-i)^{7} $$

Step-by-Step Solution

Answer

\((-1-i)^7 = -8 - 8i\)

Step 1: Convert the complex number to polar form

First, express the complex number \( -1-i \) in polar form. The modulus \( r \) is calculated as: \[ r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} \] The argument \( \theta \) is \( \tan^{-1}\left(\dfrac{-1}{-1}\right) = \tan^{-1}(1) = \frac{3\pi}{4} \), because it's in the third quadrant.

Step 2: Apply DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form, \((r(cos\theta + i \sin\theta))\), its power can be found as: \[ (r(cos\theta + i \sin\theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \]Apply it: \\((-1-i)^7 = (\sqrt{2})^7 \left( \cos\left(7 \times \frac{3\pi}{4}\right) + i \sin\left(7 \times \frac{3\pi}{4}\right) \right)\)

Step 3: Simplify the modulus

Calculate the modulus raised to the power:\[ (\sqrt{2})^7 = (\sqrt{2})^2 \times (\sqrt{2})^5 = 2^3 \times \sqrt{2} = 8\sqrt{2} \]

Step 4: Simplify the angle

Simplify the angle: \(7 \times \frac{3\pi}{4} = \frac{21\pi}{4} \).Now, express it in a simpler form by subtracting full circles \((2\pi)\):\[ \frac{21\pi}{4} - 4\pi = \frac{21\pi}{4} - \frac{16\pi}{4} = \frac{5\pi}{4} \]

Step 5: Find the trigonometric values

Now, substitute the simplified angle into the trigonometric functions: \[ (-1-i)^7 = 8\sqrt{2} (\cos(\frac{5\pi}{4}) + i \sin(\frac{5\pi}{4})) \]Since \( \frac{5\pi}{4} \) is in the third quadrant, we have \( \cos(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}} \) and \( \sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}} \).

Step 6: Multiply and find the result

Multiply the modulus by the trigonometric functions calculated in Step 5:\[ (-1-i)^7 = 8\sqrt{2} \left(-\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \]Which simplifies to: \(-8 - 8i \).

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers and are usually expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). Complex numbers provide a way to solve equations that do not have solutions among the real numbers alone, like \(x^2 + 1 = 0\). This flexibility makes them particularly useful in engineering, physics, and many areas of mathematics.

When dealing with complex numbers, it is often beneficial to convert them into a different form called the polar form.This is especially true when performing powers or roots of complex numbers, as these operations become more straightforward in polar form. Understanding how to manipulate complex numbers in both standard and polar forms is essential for using DeMoivre's Theorem effectively.

Polar Form

The polar form of a complex number represents the number in terms of its magnitude and direction. Instead of using cartesian coordinates \((a, b)\), we use polar coordinates, which consist of a distance, or modulus \(r\), and an angle, or argument \(\theta\).

The transformation from the rectangular form \(a + bi\) to the polar form \(r(\cos \theta + i\sin \theta)\) involves finding:

• **The Modulus \(r\):** It is the distance from the origin to the point \((a, b)\) in the complex plane, calculated as \(r = \sqrt{a^2 + b^2}\).
• **The Argument \(\theta\):** It is the angle formed with the positive x-axis, which is \(\theta = \tan^{-1}(b/a)\). While determining \(\theta\), it is crucial to consider the correct quadrant for the angle.

Using polar form, operations like multiplication, division, and exponentiation become easier since they primarily involve manipulating the modulus and the angle. As shown in the example, expressing \(-1-i\) in polar form makes raising it to a power simpler.

Trigonometric Functions

Trigonometric functions, specifically cosine and sine, play a vital role when dealing with complex numbers in polar form. In DeMoivre's Theorem, they allow us to manipulate the arguments of complex numbers conveniently.

When a complex number is represented as \(r(\cos \theta + i \sin \theta)\), these functions help express the number's direction:

• **Cosine (\(\cos \theta\))** gives the horizontal component (x-axis) of the angle in the complex plane.
• **Sine (\(\sin \theta\))** provides the vertical component (y-axis).

When applying DeMoivre’s Theorem to find the powers of complex numbers, we use these functions to adjust the angle:

• **Multiplying the angle** by the power \(n\), i.e., \(n\theta\).

This approach helps in simplifying the expression after the modulus is raised to the desired power. For example, converting \(7 \times \frac{3\pi}{4}\) into \(\frac{5\pi}{4}\) involves understanding the periodic nature of trigonometric functions, where cycles of \(2\pi\) represent one full circle. Thus, the trigonometric values for the same angles repeat after every \(2\pi\). This understanding is crucial when using trigonometric identities within DeMoivre's Theorem.