Problem 58 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=\sqrt{3}-i, \quad z_{2}=-\sqrt{3}-i $$

Step-by-Step Solution

Answer

\(z_{1} z_{2} = 4 \left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)\), \(\frac{z_{1}}{z_{2}} = -1\).

Step 1: Convert to Trigonometric Form

To convert each complex number to trigonometric form, calculate the magnitude and the argument of each. Start with \(z_{1} = \sqrt{3} - i\):1. Magnitude: \(|z_{1}| = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{4} = 2\).2. Argument (\(\theta_{1}\)): \(\tan^{-1} \left(\frac{-1}{\sqrt{3}}\right) = -\frac{\pi}{6}\).Thus, \(z_{1} = 2 \left(\cos\left(-\frac{\pi}{6}\right) + i\sin\left(-\frac{\pi}{6}\right)\right)\).For \(z_{2} = -\sqrt{3} - i\):1. Magnitude: \(|z_{2}| = \sqrt{(-\sqrt{3})^2 + (-1)^2} = \sqrt{4} = 2\).2. Argument (\(\theta_{2}\)): \(\tan^{-1} \left(\frac{-1}{-\sqrt{3}}\right) = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\).Thus, \(z_{2} = 2 \left(\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right)\).

Step 2: Find Product \(z_{1} \times z_{2}\)

The product of two complex numbers in trigonometric form is given by: \(z_{1} z_{2} = |z_{1}| |z_{2}| \left( \cos(\theta_{1} + \theta_{2}) + i\sin(\theta_{1} + \theta_{2}) \right)\).Here, \(|z_{1}| = 2\) and \(|z_{2}| = 2\), and the arguments are \(\theta_{1} = -\frac{\pi}{6}\) and \(\theta_{2} = \frac{5\pi}{6}\).1. Calculate the magnitude: \(|z_{1} z_{2}| = 2 \times 2 = 4\).2. Sum the angles: \(\theta_{1} + \theta_{2} = -\frac{\pi}{6} + \frac{5\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3}\).Thus, \(z_{1} z_{2} = 4 \left( \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) \right).\)

Step 3: Find Quotient \(\frac{z_{1}}{z_{2}}\)

The quotient of two complex numbers in trigonometric form is given by: \(\frac{z_{1}}{z_{2}} = \frac{|z_{1}|}{|z_{2}|} \left( \cos(\theta_{1} - \theta_{2}) + i\sin(\theta_{1} - \theta_{2}) \right)\).Here, \(|z_{1}| = 2\) and \(|z_{2}| = 2\), and the arguments are \(\theta_{1} = -\frac{\pi}{6}\) and \(\theta_{2} = \frac{5\pi}{6}\).1. Calculate the magnitude: \(\frac{|z_{1}|}{|z_{2}|} = \frac{2}{2} = 1\).2. Subtract the angles: \(\theta_{1} - \theta_{2} = -\frac{\pi}{6} - \frac{5\pi}{6} = -\frac{6\pi}{6} = -\pi\).Thus, \(\frac{z_{1}}{z_{2}} = 1 \left( \cos(-\pi) + i\sin(-\pi) \right) = -1.\)

Key Concepts

Trigonometric Form

When working with complex numbers, we often need to go beyond the standard rectangular form, which is expressed as \(a + bi\). The trigonometric form of a complex number represents the number as a point lying on the complex plane using polar coordinates. It helps simplify the multiplication and division of complex numbers.
The trigonometric form is expressed as:

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

where \(r\) represents the magnitude, and \(\theta\) is the argument.
Using the trigonometric form makes it easier to compute products and quotients due to properties of angles and magnitudes.

Magnitude and Argument

To convert a complex number like \(z = a + bi\) into its trigonometric form, you first need to determine its magnitude and argument. The magnitude \(|z|\) is the distance from the origin to the point \((a, b)\) on the complex plane, calculated using the Pythagorean theorem:

• \(|z| = \sqrt{a^2 + b^2}\)

The argument \(\theta\) is the angle between the positive real axis and the line segment from the origin to \(z\). It is found using the arctangent function:

• \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\)

Different quadrants affect \(\theta\) calculation due to angle adjustments necessary for a proper direction.

Product of Complex Numbers

To find the product of two complex numbers in trigonometric form, such as \(z_1 = r_1(\cos(\theta_1) + i\sin(\theta_1))\) and \(z_2 = r_2(\cos(\theta_2) + i\sin(\theta_2))\), you can use this simplified formula:

• \(z_1 z_2 = r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\right)\)

This method is convenient as it involves only multiplying the magnitudes \(r_1\) and \(r_2\), and adding the angles \(\theta_1\) and \(\theta_2\).
The result is a new complex number with a magnitude equal to the product of the original magnitudes and an angle that is the sum of the original angles.

Quotient of Complex Numbers

Finding the quotient of two complex numbers is also more straightforward in trigonometric form. For two complex numbers \( z_1\) and \(z_2\), the quotient is given by:

• \(\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\right)\)

This method only requires dividing the magnitudes \(r_1\) and \(r_2\) and subtracting the arguments \(\theta_1\) and \(\theta_2\).
The result is a complex number with a magnitude equal to the quotient of the original magnitudes and an angle that is the difference of the original angles.