Problem 24 - Precalculus

Question

Find the result of each expression using De Moivre's theorem. Write the answer in rectangular form. $$(\sqrt{3}-i)^{8}$$

Step-by-Step Solution

Answer

The result is \\(-128 + 128\sqrt{3}i\\) in rectangular form.

Step 1: Convert to Polar Form

To apply De Moivre's theorem, we first convert the complex number \(\sqrt{3} - i\) into polar form. The modulus is calculated as \(|z| = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = 2\). The argument \((\theta)\) is found using \((\theta = \arctan\left(\frac{-1}{\sqrt{3}}\right) = -\frac{\pi}{6}\). Multiplying both the modulus and the computed argument by 8 for the power transform yields radius \(2^8\) and angle \(-\frac{8\pi}{6} = -\frac{4\pi}{3}\).

Step 2: Apply De Moivre's Theorem

De Moivre's theorem states that \( (r(\cos \theta + i\sin \theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\). Using this with our transformed angle and radius, we get: \[\left(2\right)^8 \left( \cos\left(-\frac{4\pi}{3}\right) + i\sin\left(-\frac{4\pi}{3}\right) \right)\].

Step 3: Calculate Power of Radius

Compute the power of the modulus: \(2^8 = 256\). We now express the formula as \[256(\cos\left(-\frac{4\pi}{3}\right) + i\sin\left(-\frac{4\pi}{3}\right))\].

Step 4: Evaluate Trigonometric Functions

We calculate \( \cos(-\frac{4\pi}{3}) = -\frac{1}{2} \) and \( \sin(-\frac{4\pi}{3}) = \frac{\sqrt{3}}{2} \). Substitute these into the expression: \[256\left(-\frac{1}{2} + i\frac{\sqrt{3}}{2} \right)\].

Step 5: Simplify to Rectangular Form

Multiply 256 by each component: \[256 \times -\frac{1}{2} = -128\quad\text{and}\quad256 \times \frac{\sqrt{3}}{2} = 128\sqrt{3}\]. Therefore, in rectangular form, the result is \(-128 + 128\sqrt{3}i\).

Key Concepts

Polar Form

The polar form of a complex number is a way of expressing it by using its magnitude and direction (angle) rather than its real and imaginary parts.
This form is especially useful in complex analysis and when dealing with multiplication and powers of complex numbers. In the exercise above, we converted the complex number \( \sqrt{3} - i \) into polar form to simplify the application of De Moivre’s theorem.

To convert a complex number to polar form:

• Calculate the modulus (or absolute value), given by \( |z| = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the real and imaginary parts, respectively.
• Determine the argument \( \theta \), by calculating \( \tan^{-1}(b/a) \).

For \( \sqrt{3} - i \), we found the modulus to be 2 and the argument to be \(-\frac{\pi}{6}\). These parameters form our polar expression: \(2(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6}))\).

Complex Numbers

Complex numbers extend the concept of one-dimensional numbers to the two-dimensional complex plane by adding an 'imaginary' axis perpendicular to the 'real' axis.
They take the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part, \( i \) being the imaginary unit.

These numbers are critical for many fields, including engineering and physics, as they can represent waveforms, oscillations, and electrical circuits, among other things. In the current problem, \( \sqrt{3} - i \) is presented as a complex number, where \( \sqrt{3} \) is the real part, and \(-1i\) is the imaginary part. By understanding complex numbers, we are better equipped to manipulate expressions like this and solve problems using powerful tools like De Moivre’s theorem.

Rectangular Form

The rectangular form of a complex number is its most familiar expression, \( a + bi \). This form directly indicates the real and imaginary components.

• The 'real part' aligns with the x-axis (on a standard graph), and the imaginary part aligns with the y-axis.
• The form is efficient for addition, subtraction, and straightforward arithmetic operations.

Returning complex numbers back to rectangular form after calculations in polar form is often useful for practical interpretations.
For example, in the given exercise, after using De Moivre's theorem in polar form, the final polar results are converted back to rectangular form following the formula \(-128 + 128\sqrt{3}i\). This final format is generally preferred in many applications for its simplicity and direct readability.

Trigonometric Functions

Trigonometric functions such as cosine and sine are fundamental to working with complex numbers in polar form.
They relate angles to side lengths in right-angled triangles, forming the backbone of circular and spherical mathematics.
In the problem, after converting to polar form and realizing that De Moivre's power was being applied, we computed:

• \( \cos(-\frac{4\pi}{3}) = -\frac{1}{2} \)
• \( \sin(-\frac{4\pi}{3}) = \frac{\sqrt{3}}{2} \)

These calculations are crucial as they translate the polar back to rectangular form by scaling these values with the modulus.
Understanding these trigonometric functions enables the transformation between polar and rectangular forms and is a cornerstone in handling complex numbers.