Problem 51 - 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. $$ \frac{w^{2}}{z^{3}} $$

Step-by-Step Solution

Answer

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

Step 1: Express Complex Numbers in Polar Form

First, express the given complex numbers \(z\) and \(w\) in polar form. For number \(z = -\frac{3 \sqrt{3}}{2} + \frac{3}{2} i\), calculate the magnitude \(r_z = \sqrt{(-\frac{3 \sqrt{3}}{2})^2 + (\frac{3}{2})^2}\) and the argument \(\theta_z = \text{atan2}(\frac{3}{2}, -\frac{3 \sqrt{3}}{2})\). Similarly, for \(w = 3 \sqrt{2} - 3 i \sqrt{2}\), find \(r_w = \sqrt{(3 \sqrt{2})^2 + (-3 \sqrt{2})^2}\) and \(\theta_w = \text{atan2}(-3 \sqrt{2}, 3 \sqrt{2})\).

Step 2: Calculate Magnitude and Argument for z

Calculate \(r_z = \sqrt{\left(-\frac{3\sqrt{3}}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \sqrt{\frac{27}{4} + \frac{9}{4}} = \sqrt{9} = 3\). The argument \(\theta_z = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\). So, \(z = 3\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right)\).

Step 3: Calculate Magnitude and Argument for w

Calculate \(r_w = \sqrt{(3\sqrt{2})^2 + (-3\sqrt{2})^2} = \sqrt{18 + 18} = \sqrt{36} = 6\). The argument \(\theta_w = \text{atan2}(-3\sqrt{2}, 3\sqrt{2}) = -\frac{\pi}{4}\). So, \(w = 6\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\).

Step 4: Calculate w^2 in Polar Form

To find \(w^2\), use \(r_w^2 = 6^2 = 36\) and \(2\theta_w = 2\left(-\frac{\pi}{4}\right) = -\frac{\pi}{2}\). Hence, \(w^2 = 36 \left(\cos\left(-\frac{\pi}{2}\right) + i\sin\left(-\frac{\pi}{2}\right)\right)\).

Step 5: Calculate z^3 in Polar Form

To find \(z^3\), calculate \(r_z^3 = 3^3 = 27\) and \(3\theta_z = 3\times\frac{5\pi}{6} = \frac{15\pi}{6} = \frac{5\pi}{2}\). Taking the principal value, it becomes \(\frac{\pi}{2}\). Thus, \(z^3 = 27\left(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\right)\).

Step 6: Divide the Two Quantities

Now calculate \(\frac{w^2}{z^3}\). The magnitude is \(\frac{36}{27} = \frac{4}{3}\), and the argument is \(-\frac{\pi}{2} - \frac{\pi}{2} = -\pi\). Thus, \(\frac{w^2}{z^3} = \frac{4}{3} \left(\cos(-\pi) + i\sin(-\pi)\right)\) in polar form.

Key Concepts

Polar Form

When talking about complex numbers, expressing them in polar form is an alternative to the common rectangular (or Cartesian) form. This provides a way to represent complex numbers based on their magnitude and angle relative to the real axis. In polar form, a complex number is written as:

• \( z = r(\cos \theta + i\sin \theta) \)

Here, \( r \) denotes the magnitude (or modulus) of the complex number, while \( \theta \), known as the argument, specifies the direction. Rather than relying on the real part \( a \) and the imaginary part \( b \) individually, polar form allows a more geometrically intuitive representation.
The process for converting a complex number from rectangular to polar form involves calculating these components:

• Magnitude: \( r = \sqrt{a^2 + b^2} \)
• Argument: \( \theta = \text{atan2}(b, a) \)

This approach is particularly useful for multiplying or dividing complex numbers, as well as raising them to powers, due to the straightforward operations involving their magnitudes and arguments.

Principal Argument

The principal argument of a complex number is its argument \( \theta \) restricted within a specific range. Typically, this range is between \(-\pi\) and \(\pi\). This ensures a unique representation of any angle, as angles in mathematics are inherently periodic. Hence, only a single key angle needs to be referred to when dealing with complex numbers.
The expression for the principal argument comes from the two-argument arctangent function, often denoted as \( \text{atan2} \). It evaluates the appropriate quadrant for \( \theta \) based on the signs and values of \( a \) and \( b \), where the complex number is written as \( a + bi \).
By ensuring that the argument is confined to the principal range, it becomes easier to communicate and compute consistently. Functions like sine and cosine, which are periodic, lead to multiple angles having similar representations, thus noting the primary angle or principal argument retains clarity in complex number operations.

Magnitude and Argument

The magnitude and argument are two fundamental aspects that fully describe a complex number when expressed in polar form. To start with, the magnitude \( r \) is a non-negative number that represents the distance from the origin of the complex plane (the point \( (0,0) \)) to the point representing the complex number \( z \). Hence, it is always positive or zero because it is derived from the Pythagorean theorem.
Calculating the magnitude involves:

• Taking the square root of the sum of the squares of the real part and the imaginary part: \( r = \sqrt{a^2 + b^2} \)

The argument of a complex number, denoted as \( \theta \), is the angle made with the positive real axis. This angle is measured in radians and can take on values within its principal range. It dictates the direction from the origin to the point in the complex plane.
Together, the magnitude and argument rolled into the polar coordinate system offer an insightful way to grasp the orientation and length of vectors associated with complex quantum states or signal processes in engineering. The ability to calculate these effortlessly allows one to switch representations of the complex number, particularly useful in mathematical operations involving complex numbers such as multiplication, division, and raising to powers.