Problem 60 - College Trigonometry

Question

Use DeMoivre's Theorem to find the indicated power of the given complex number. Express your final answers in rectangular form. $$ \left(\frac{\sqrt{3}}{3}-\frac{1}{3} i\right)^{4} $$

Step-by-Step Solution

Answer

The rectangular form is \(-\frac{8}{81} - \frac{8\sqrt{3}}{81}i\).

Step 1: Express the Complex Number in Polar Form

The given complex number is \( \frac{\sqrt{3}}{3} - \frac{1}{3} i \). First, find its magnitude \( r = \sqrt{\left(\frac{\sqrt{3}}{3}\right)^2 + \left(-\frac{1}{3}\right)^2} = \sqrt{\frac{1}{3} + \frac{1}{9}} = \sqrt{\frac{4}{9}} = \frac{2}{3} \). Next, find the argument (angle) using \( \theta = \tan^{-1} \left(\frac{-\frac{1}{3}}{\frac{\sqrt{3}}{3}}\right) = \tan^{-1}(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6} \) radians. Thus, our complex number in polar form is \( \frac{2}{3} \text{cis}(-\frac{\pi}{6}) \).

Step 2: Apply DeMoivre's Theorem

Using DeMoivre's Theorem, \( \left(r \text{cis}(\theta)\right)^n = r^n \text{cis}(n\theta) \). For \( n = 4 \), and \( r = \frac{2}{3} \), \( \theta = -\frac{\pi}{6} \), we have: \[ \left( \frac{2}{3} \right)^4 \text{cis} \left( 4 \times -\frac{\pi}{6} \right) \] Simplifying, \( \left( \frac{2}{3} \right)^4 = \frac{16}{81} \) and the argument is \(-\frac{4\pi}{6} = -\frac{2\pi}{3} \). Thus, the complex number becomes \( \frac{16}{81} \text{cis}(-\frac{2\pi}{3}) \).

Step 3: Convert Back to Rectangular Form

The polar form \( \text{cis}(-\frac{2\pi}{3}) \) corresponds to: \( \cos(-\frac{2\pi}{3}) = -\frac{1}{2} \) and \( \sin(-\frac{2\pi}{3}) = -\frac{\sqrt{3}}{2} \). Hence, the rectangular form is: \[ \frac{16}{81} \left( \cos(-\frac{2\pi}{3}) + i \sin(-\frac{2\pi}{3}) \right) = \frac{16}{81} \left(-\frac{1}{2} - \frac{\sqrt{3}}{2}i\right) \] Multiply by \( \frac{16}{81} \): \[ -\frac{8}{81} - \frac{8\sqrt{3}}{81}i \].

Key Concepts

Polar Form of Complex Numbers

Complex numbers can be expressed in different forms, including the polar form, which is especially useful for multiplication and division. A complex number in its polar form is written as \( r \text{cis}(\theta) \), where:

• \( r \) is the magnitude (or modulus) of the complex number.
• \( \theta \) is the argument (or angle) measured in radians.
• "cis" is shorthand for \( \cos(\theta) + i\sin(\theta) \).

To find the polar form, begin with calculating the magnitude \( r \) using the formula \( r = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the real and imaginary parts of the complex number, respectively. This gives you the "size" of the number on the complex plane.
Next, determine the argument \( \theta \) using an inverse tangent or appropriate trigonometric functions, often expressed as \( \tan^{-1}\left(\frac{y}{x}\right) \). The polar form transforms the complex number into something easier to manipulate with mathematical operations like powering.

Rectangular Form of Complex Numbers

The rectangular form of a complex number is often the most straightforward way to express it. It is written as \( a + bi \), where:

• \( a \) is the real part of the complex number.
• \( b \) is the imaginary part.

This form is often referred to as the Cartesian form because it represents numbers in a 2D plane with real and imaginary axes.
Transitioning between rectangular and polar forms requires algebraic manipulation. To revert a polar form \( r \text{cis}(\theta) \) back to rectangular, use:

• The cosine of the angle \( \theta \) for the real part: \( x = r \cos(\theta) \).
• The sine of the angle \( \theta \) for the imaginary part: \( y = r \sin(\theta) \).

This allows us to express any complex number in its usual \( a + bi \) form, which makes it easier to perform addition and subtraction.

Argument of a Complex Number

The argument of a complex number is the angle \( \theta \) that defines the direction of the number on the complex plane. It indicates how much the imaginary part "turns" from the real axis, which crucially affects the number's behavior when multiplied or raised to a power.
The argument \( \theta \) is generally found with the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). This gives the principal value of the argument in radians, usually within the range \(-\pi\) to \(\pi\). Understanding this angle is vital because:

• It dictates the rotation of the number around the origin of the complex plane.
• It influences the result when applying DeMoivre's Theorem, allowing us to multiply or take powers of complex numbers efficiently.

By quantifying the angle, we can convert between forms and perform complex operations easily.