Problem 64 - College Trigonometry

Question

Use DeMoivre's Theorem to find the indicated power of the given complex number. Express your final answers in rectangular form. $$ (1-i)^{8} $$

Step-by-Step Solution

Answer

The result is \(16\) in rectangular form.

Step 1: Express the Complex Number in Polar Form

Given the complex number \(1 - i\), we need to convert it into polar form. The modulus \(r\) is calculated as \( \sqrt{1^2 + (-1)^2} = \sqrt{2}\). The argument \(\theta\) can be found using \( \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4}\). Thus, \(1-i\) in polar form is \( \sqrt{2} \text{cis}(-\frac{\pi}{4})\), where \(\text{cis}\theta = \cos\theta + i\sin\theta\).

Step 2: Apply DeMoivre's Theorem

DeMoivre's Theorem states that \((r\cdot\text{cis}\theta)^n = r^n\cdot\text{cis}(n\theta)\). For \(n=8\), we have:\[ (1-i)^8 = (\sqrt{2}\cdot\text{cis}(-\frac{\pi}{4}))^8 = (\sqrt{2})^8 \cdot \text{cis}(-8\cdot\frac{\pi}{4}).\]Calculating gives \((\sqrt{2})^8 = 16\) and \(-8\cdot\frac{\pi}{4} = -2\pi\).

Step 3: Simplify the Polar Form Result

Now compute \(16 \cdot \text{cis}(-2\pi)\). The angle \(-2\pi\) is equivalent to \(0\) (since it's one full rotation around the circle). Thus, \(\text{cis}(-2\pi) = 1\). Therefore, the expression simplifies to \(16\cdot 1 = 16\).

Step 4: Express the Result in Rectangular Form

The polar expression \(16\) represents \(16 + 0i\) in rectangular form. Thus, the rectangular form of the complex number \((1-i)^8\) is simply \(16\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They are typically written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with \(i\) representing the imaginary unit, satisfying \(i^2 = -1\). The beauty of complex numbers is their ability to express numbers that are not easily represented on the real number line. For instance, the complex number \(1-i\) has a real part, 1, and an imaginary part, -1i.

• Real Part: this is the part without the imaginary unit, i.e., \(a\)
• Imaginary Part: this is the coefficient of \(i\), i.e., \(b\) in \(bi\)

By using complex numbers, we can solve equations that have no solutions in the set of real numbers, such as quadratic equations with negative discriminants. They offer a way to graph numbers beyond the one-dimensional number line.

Polar Form

The polar form of a complex number expresses it in terms of a modulus and an argument, providing an alternative way of representing complex numbers compared to the rectangular form. In polar form, a complex number is expressed as \(r\text{cis}\theta\), where \(r\) is the modulus and \(\theta\) is the argument. This form is particularly useful when multiplying or dividing complex numbers, or when raising them to a power.

• \(r\) (modulus): magnitude of the complex number
• \(\theta\) (argument): angle with the positive real axis
• Form: \(r(\cos\theta + i\sin\theta)\), simplified to \(r\text{cis}\theta\)

By converting \(1-i\) to polar form, we found \(\sqrt{2}\text{cis}(-\frac{\pi}{4})\). This makes operations like exponentiation straightforward and simple to perform.

Rectangular Form

The rectangular form of a complex number is the standard way we write them as \(a+bi\). This form emphasizes the real and imaginary parts separately, which is very intuitive for addition and subtraction of complex numbers. Unlike the polar form, the rectangular form doesn’t inherently carry magnitude or direction, but shows the "coordinates" of a complex number on a 2D plane.

• \(a\): the real part
• \(b\): the imaginary part

Calculations like \(1-i\) raised to the 8th power initially require conversion to polar form, but once computations are completed in polar form, the result can be easily converted back. The final outcome of \((1-i)^8 = 16 + 0i\) directly shows the real number gained through these operations.

Modulus

The modulus of a complex number is essentially its length or distance from the origin on the complex plane. It is denoted by \(r\) and calculated using the Pythagorean theorem, \(r = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts, respectively. The modulus is always non-negative and it provides a measure of the magnitude of the complex number.

• Finding \(r\): \(\sqrt{1^2 + (-1)^2} = \sqrt{2}\)

For \(1-i\), the modulus \(\sqrt{2}\) helps us transition to polar form and enables efficient calculations when used in combination with formulas like DeMoivre's Theorem.

Argument

The argument of a complex number is the angle it makes with the positive real axis. It is typically denoted by \(\theta\) and is crucial in converting a complex number to its polar form. The argument is determined from the complex number's position, calculated using \(\tan^{-1}(\frac{b}{a})\), where \(b\) and \(a\) are the imaginary and real parts, respectively.

• Calculation: For \(1-i\), \(\tan^{-1}(\frac{-1}{1}) = -\frac{\pi}{4}\)
• Range: Often expressed in radians, and usually within the interval (-\(\pi\), \(\pi\)]

The argument allows us to efficiently work with complex numbers in polar form. Understanding the argument is key, as it determines the rotational aspect of the number on the plane.