Problem 44 - College Trigonometry

Question

Use \(z=-\frac{3 \sqrt{3}}{2}+\frac{3}{2} i\) and \(w=3 \sqrt{2}-3 i \sqrt{2}\) to compute the quantity. Express your answers in polar form using the principal argument. $$ z^{4} $$

Step-by-Step Solution

Answer

\(z^4 = 81 \left( \cos\left(-\frac{2\pi}{3}\right) + i \sin\left(-\frac{2\pi}{3}\right) \right)\).

Step 1: Convert to Polar Form

The first step is to convert the complex number \(z = -\frac{3\sqrt{3}}{2} + \frac{3}{2} i\) into polar form. Start by finding the modulus: \(|z| = \sqrt{\left(-\frac{3\sqrt{3}}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \sqrt{\frac{27}{4} + \frac{9}{4}} = \sqrt{\frac{36}{4}} = \sqrt{9} = 3\). Next, find the argument: Compute \(\theta = \tan^{-1} \left( \frac{3/2}{-3\sqrt{3}/2} \right) = \tan^{-1} \left(-\frac{1}{\sqrt{3}}\right)\). This gives \(\theta = -\frac{\pi}{6}\). Thus, in polar form, \(z = 3 \left( \cos\left(-\frac{\pi}{6} \right) + i \sin\left(-\frac{\pi}{6} \right) \right)\).

Step 2: Calculate \(z^4\) in Polar Form

Use the polar form from Step 1 to compute \(z^4\). If \(z = r(\cos \theta + i \sin \theta)\), then \(z^4 = r^4 \left( \cos(4\theta) + i \sin(4\theta) \right)\).Given \(r = 3\) and \(\theta = -\frac{\pi}{6}\), calculate the modulus: \(r^4 = 3^4 = 81\).Now compute the argument: \(4\theta = 4 \times -\frac{\pi}{6} = -\frac{2\pi}{3}\). Thus, \(z^4 = 81 \left( \cos\left(-\frac{2\pi}{3}\right) + i \sin\left(-\frac{2\pi}{3}\right) \right)\).

Step 3: Simplify the Result

To express \(z^4\) in standard or polar form, compute \(\cos\left(-\frac{2\pi}{3}\right)\) and \(\sin\left(-\frac{2\pi}{3}\right)\), which are \(-\frac{1}{2}\) and \(-\frac{\sqrt{3}}{2}\) respectively. Therefore, \(z^4 = 81 \left(-\frac{1}{2} + i\left(-\frac{\sqrt{3}}{2}\right)\right) = 81 \left(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\right)\).Hence, \(z^4 = -\frac{81}{2} - i\frac{81\sqrt{3}}{2}\). In polar form, it remains \(z^4 = 81 \left( \cos\left(-\frac{2\pi}{3}\right) + i \sin\left(-\frac{2\pi}{3}\right) \right)\).

Key Concepts

Polar Form

In complex numbers, every complex number can be expressed in a form that highlights its geometric properties. This is known as the polar form. To represent a complex number in polar form, it is characterized by two components: the modulus and the argument. The polar form of a complex number is given as:

• For a complex number \[ z = a + bi \] its polar form is: \[ z = r(\cos \theta + i \sin \theta) \] where \
• \( r \) is the modulus, the distance from the origin to the point in the complex plane.
• \( \theta \) is the argument, the angle formed with the positive real axis.

The advantage of using polar form comes when you are performing multiplication, division, and raising to powers. In polar form, a complex number is an elegant and intuitive way to see how the magnitude and direction interact.
Here, using this form simplifies calculations significantly, especially when applying powers to a complex number.

Modulus

The modulus of a complex number is a fundamental concept that measures its "length" in the complex plane. For any complex number \( z = a + bi \) the modulus, denoted as \( |z| \) is calculated using the formula:

• \[ |z| = \sqrt{a^2 + b^2} \]

This formula results from the Pythagorean theorem, providing the hypotenuse (modulus) for the right triangle formed in the complex plane.
In simpler terms, imagine the complex number as a point on a 2D grid; the modulus tells you how far that point is from the origin \((0, 0)\).

• In the exercise, \[ z = -\frac{3\sqrt{3}}{2} + \frac{3}{2} i \] which calculates to \(|z| = 3\).

A larger modulus means the point is farther from the origin, while a smaller modulus means it's closer. This measure of distance is crucial, especially when multiplying complex numbers, as it directly affects the operation outcome.

Argument

The argument of a complex number is another key concept that describes the angle's direction in the complex plane. It is the angle formed between the positive real axis and the line segment joining the origin and the point representing the complex number. If we consider a complex number \(z = a + bi\),

• the argument \(\theta\) is given by \[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \].

This angle helps in understanding how the complex number is oriented spatially. The argument can take positive or negative values depending on the quadrant where the complex number resides.
For the exercise, the argument was:

• \(\theta = -\frac{\pi}{6}\).

Remember, arguments are very helpful when visualizing complex numbers on the Argand plane. A positive argument angle indicates a direction counter-clockwise from the positive real axis, whereas a negative value indicates clockwise.

De Moivre's Theorem

De Moivre's Theorem is a powerful tool used in complex arithmetic, especially for raising powers of complex numbers in their polar form. The theorem states:

• For any complex number \( z = r (\cos \theta + i \sin \theta) \) and integer \( n \), \( z^n = r^n (\cos(n \theta) + i \sin(n \theta)) \).

This converts the problem of powers into simpler arithmetic. De Moivre’s theorem uses the properties of trigonometric functions to simplify power calculations drastically. In our exercise:

• We had: \( z^4 = 81 \left( \cos\left(-\frac{2\pi}{3}\right) + i \sin\left(-\frac{2\pi}{3}\right) \right) \).

By applying De Moivre's Theorem, we calculate powers of \(z\) with ease by multiplying the argument by the power and raising the modulus to that power. It is a remarkably efficient method for handling powers, especially compared to traditional algebraic expansion.