Problem 77 - Algebra and Trigonometry

Question

Find the indicated power using De Moivre's Theorem. $$ (-1-i)^{7} $$

Step-by-Step Solution

Answer

The result is \(-4\sqrt{2} + 4i\sqrt{2}\).

Step 1: Convert to Polar Form

First, we need to express the complex number \(-1-i\) in polar form. To do this, find the modulus and argument of the complex number.- **Modulus**: \( r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2} \).- **Argument**: \( \theta = \tan^{-1}\left(\frac{-1}{-1}\right) = -\frac{3\pi}{4} \). Thus, the polar form is: \[ -1-i = \sqrt{2} \left( \cos\left(-\frac{3\pi}{4}\right) + i\sin\left(-\frac{3\pi}{4}\right) \right) \].

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form, \((r(\cos\theta + i\sin\theta))^n = r^n (\cos(n\theta) + i\sin(n\theta))\). Apply this to find \((-1-i)^7\):- \( r^7 = (\sqrt{2})^7 = 2^{7/2} \).- \( 7 \times (-\frac{3\pi}{4}) = -\frac{21\pi}{4} \).Use cosine and sine identities to simplify: this becomes the equivalent angle \(-\frac{21\pi}{4} = -\frac{5\pi}{4}\) (adding \(2\pi\) twice to bring it within standard interval).

Step 3: Simplify Using Trigonometric Values

Calculate the trigonometric values:- \( \cos\left(-\frac{5\pi}{4}\right) = -\frac{1}{\sqrt{2}} \).- \( \sin\left(-\frac{5\pi}{4}\right) = \frac{1}{\sqrt{2}} \).Substitute these into De Moivre’s Theorem outcome from step 2: \[ 2^{7/2} \left( -\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \right) = 4\sqrt{2}(-1 + i) \].

Step 4: Final Expression in Rectangular Form

Express in the standard rectangular form by evaluating:Multiply out: - \( 4\sqrt{2} \times -1 = -4\sqrt{2} \).- \( 4\sqrt{2} \times i = 4i\sqrt{2} \).Combine to form: \(-4\sqrt{2} + 4i\sqrt{2}\).

Key Concepts

Complex Numbers

Complex numbers are like a combination of real numbers and imaginary numbers. They live on a plane, called the complex plane, which makes them quite different from the number line we use for real numbers.
Any complex number can be written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here, \(b\) is a real number, and \(i\) is the imaginary unit with the property that \(i^2 = -1\).
For example, the complex number \(-1-i\) has a real part of -1 and an imaginary part of -1 as well.

• Real part: \(a\)
• Imaginary part: \(bi\)

Understanding complex numbers starts with knowing how to add, subtract, multiply, and divide them, similar to algebra. They are crucial in fields like engineering and physics, providing essential ways of describing waves and oscillations.

Polar Form

The polar form of a complex number is another way to express it, especially useful for multiplication and finding powers. Instead of using real and imaginary parts, we use magnitude (or modulus) and angle (or argument). This way, they look much like vectors.
For a complex number \(a + bi\), its modulus \(r\) is defined as \(r = \sqrt{a^2 + b^2}\). The argument \(\theta\) is the angle made with the positive x-axis and can be found using \(\theta = \tan^{-1}(\frac{b}{a})\).
Thus, the polar form is \(r(\cos\theta + i\sin\theta)\). This representation makes it easier to apply De Moivre's theorem, which is used to calculate powers of complex numbers.

• Modulus: \(r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}\)
• Argument: \(\theta = \tan^{-1}(\frac{-1}{-1}) = -\frac{3\pi}{4}\)

Converting \(-1-i\) into polar form gives us \(\sqrt{2} (\cos(-\frac{3\pi}{4}) + i\sin(-\frac{3\pi}{4}))\).

Trigonometric Identities

Trigonometric identities help simplify calculations involving sine and cosine, especially when working with angles beyond usual range. They are vital in converting complex numbers between forms.
For De Moivre's Theorem, which involves raising complex numbers to a power, knowing the basic trigonometric functions of specific angles is crucial. Consider the identity adjustment made for periodicity: reducing \(-\frac{21\pi}{4}\) to \(-\frac{5\pi}{4}\) by adding \(2\pi\) appropriately.

• Cosine: \(\cos(-\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}\)
• Sine: \(\sin(-\frac{5\pi}{4}) = \frac{1}{\sqrt{2}}\)

These values help in simplifying the expression using De Moivre’s Theorem, showcasing how trigonometric identities play a role.

Powers of Complex Numbers

When you need to raise a complex number to a power, De Moivre's Theorem is your best friend. This theorem states that for any complex number in polar form, \((r(\cos\theta + i\sin\theta))^n = r^n (\cos(n\theta) + i\sin(n\theta))\). It's a powerful tool for simplifying what might seem complicated.
For the complex number \(-1-i\), first express it in polar form, then apply the theorem. You’ll simplify the problem by raising the modulus to the power and multiplying the angle by the power.
After applying the theorem, substitution of trigonometric identities allows reconversion to rectangular form. For the example \((-1-i)^7\), this leads to the final result:

• Polar form power: \(2^{7/2} (\cos(-\frac{5\pi}{4}) + i\sin(-\frac{5\pi}{4}))\)
• Rectangular form: \(-4\sqrt{2} + 4i\sqrt{2}\)

Understanding this process makes it easy to handle complex numbers raised to any power!