Problem 7 - Precalculus

Question

Find the product \(z_{1} z_{2}\) and express it in rectangular form. $$z_{1}=\sqrt{3}\left[\cos \left(\frac{\pi}{12}\right)+i \sin \left(\frac{\pi}{12}\right)\right] \text { and } z_{2}=\sqrt{27}\left[\cos \left(\frac{\pi}{6}\right)+i \sin \left(\frac{\pi}{6}\right)\right]$$

Step-by-Step Solution

Answer

The product \( z_1 z_2 \) in rectangular form is \( \frac{9\sqrt{2}}{2} + i \frac{9\sqrt{2}}{2} \).

Step 1: Express Complex Numbers in Polar Form

The complex numbers given are already expressed in polar form. - \( z_1 = \sqrt{3} \left[ \cos\left( \frac{\pi}{12} \right) + i \sin\left( \frac{\pi}{12} \right) \right] \)- \( z_2 = \sqrt{27} \left[ \cos\left( \frac{\pi}{6} \right) + i \sin\left( \frac{\pi}{6} \right) \right] \)

Step 2: Use the Formula for Multiplying Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, the moduli are multiplied and the angles are added:- \( |z_1 z_2| = |z_1| \cdot |z_2| \)- \( \text{arg}(z_1 z_2) = \text{arg}(z_1) + \text{arg}(z_2) \)Substitute the values:\(|z_1| = \sqrt{3}, \quad |z_2| = \sqrt{27} = 3\sqrt{3},\)\(\text{arg}(z_1 z_2) = \frac{\pi}{12} + \frac{\pi}{6} = \frac{\pi}{12} + \frac{2\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4}.\)

Step 3: Calculate the Product in Polar Form

Multiply the moduli:\[|z_1 z_2| = \sqrt{3} \times 3\sqrt{3} = 9\]Therefore, the expression for \( z_1 z_2 \) in polar form is:\[z_1 z_2 = 9 \left[ \cos\left( \frac{\pi}{4} \right) + i \sin\left( \frac{\pi}{4} \right) \right]\]

Step 4: Convert to Rectangular Form

To convert from polar to rectangular form, use the trigonometric identities for \( \cos \) and \( \sin \) at angle \( \frac{\pi}{4} \):\[\cos\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}, \, \sin\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}\]Now calculate:\[z_1 z_2 = 9 \left( \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right)\]Distribute the 9:\[z_1 z_2 = 9 \times \frac{\sqrt{2}}{2} + 9i \times \frac{\sqrt{2}}{2}\]Simplify:\[z_1 z_2 = \frac{9\sqrt{2}}{2} + i \frac{9\sqrt{2}}{2}\]

Key Concepts

Polar Form

Complex numbers in polar form are a way to express these numbers using a radius and an angle. They take the format \(z = r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude of the complex number and \(\theta\) is the angle, or the argument, on the complex plane.
Think of \(r\) as the distance from the origin to a point, and \(\theta\) as the angle that line makes with the positive x-axis. This form is particularly useful when multiplying or dividing complex numbers, as it simplifies those operations.
In the given exercise, the numbers \(z_1\) and \(z_2\) are presented in polar form, which shows us their magnitude and angle immediately without needing additional calculations.

Rectangular Form

Rectangular form, also known as Cartesian form, represents complex numbers as \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
This form is more intuitive for performing addition and subtraction with complex numbers, as it straightforwardly separates their real and imaginary components.
To convert from polar to rectangular form, we use the trigonometric identities: \(\cos \theta = \frac{x}{r}\) and \(\sin \theta = \frac{y}{r}\), leading us to \(x = r \cos \theta\) and \(y = r \sin \theta\).
For the product of \(z_1 z_2\), after finding it in polar form, we switched to rectangular form to express the real and imaginary components separately, offering a geometric understanding of its position in the complex plane.

Multiplication of Complex Numbers

Multiplying complex numbers is straightforward in polar form. You multiply the magnitudes (or lengths) and add the angles.
The formula \( |z_1 z_2| = |z_1| \cdot |z_2| \) and \( \text{arg}(z_1 z_2) = \text{arg}(z_1) + \text{arg}(z_2) \) reflects this process.
For example, in the exercise, multiplying \(z_1\) and \(z_2\) means calculating the new magnitude as \(\sqrt{3} \times 3\sqrt{3} = 9\), and the new angle as \(\frac{\pi}{12} + \frac{\pi}{6} = \frac{\pi}{4}\).
This step, simply put, combines their rotational and distance properties into a single new complex number, significantly speeding up the operation compared to rectangular form.

Trigonometric Identities

Trigonometric identities are crucial when dealing with complex numbers, especially during conversion between forms.
They provide the relationship between angles and side lengths in a right triangle, which can be leveraged in both polar and rectangular forms.
For the calculation, it's important to remember that \(\cos\) and \(\sin\) of \(\frac{\pi}{4}\) are both \(\frac{\sqrt{2}}{2}\), which simplifies multiplying and converting steps.
Understanding these identities means you can predict the impact of angle changes on the real and imaginary parts of a complex number, enabling accurate calculations and conversions between the forms.