Problem 45 - Algebra and Trigonometry

Question

In Exercises \(45-52,\) find the quotient \(\frac{z_{1}}{z_{2}}\) of the complex numbers. Leave answers in polar form. In Exercises \(49-50,\) express the argument as an angle between \(0^{\circ}\) and \(360^{\circ}\). $$ \begin{array}{l} {z_{1}=20\left(\cos 75^{\circ}+i \sin 75^{\circ}\right)} \\ {z_{2}=4\left(\cos 25^{\circ}+i \sin 25^{\circ}\right)} \end{array} $$

Step-by-Step Solution

Answer

The quotient of the given complex numbers in polar form is \(5(\cos 50^{\circ} + i \sin 50^{\circ})\)

Step 1: Identify the magnitudes and arguments

Identify the magnitudes \(r_{1}\) and \(r_{2}\) and arguments \(\theta_{1}\) and \(\theta_{2}\) of the provided complex numbers. Here, \(r_{1} = 20, \theta_{1} = 75^{\circ}, r_{2} = 4, \theta_{2} = 25^{\circ}\)

Step 2: Compute the magnitude of the quotient

Calculate the magnitude of the quotient by dividing \(r_{1}\) by \(r_{2}\). Thus, the magnitude of the quotient is \(20/4 = 5\)

Step 3: Compute the argument of the quotient

The argument of the quotient is the difference \(\theta_{1} - \theta_{2}\), subtract \(25^{\circ}\) from \(75^{\circ}\) giving \(50^{\circ}\). This value is already between 0 and 360 degrees so it does not need modification.

Step 4: Formulate the quotient in polar form

So, the quotient in polar form is \(5(\cos 50^{\circ} + i \sin 50^{\circ})\)

Key Concepts

Polar Form

Representing complex numbers in polar form is a great way to simplify many operations, including multiplication and division. In polar form, a complex number is expressed as \[r(\cos \theta + i\sin \theta)\] where:

• \(r\) is the magnitude of the complex number, which represents its distance from the origin in the complex plane.
• \(\theta\) is the argument, which is the angle in degrees or radians the line makes with the positive x-axis.

When a complex number is in polar form, it provides a much clearer insight into the geometric representation and behaviors during operations like rotation and scaling. Converting a complex number into polar form involves calculating both its magnitude and argument.

Quotient of Complex Numbers

When dividing two complex numbers, it is often advantageous to convert them into their polar forms. This simplifies the process significantly. The procedure to find the quotient \(\frac{z_1}{z_2}\) of two complex numbers \(z_1\) and \(z_2\) is straightforward:

• Divide the magnitudes: If \(z_1 = r_1(\cos \theta_1 + i\sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i\sin \theta_2)\), then the magnitude of the quotient is simply \(\frac{r_1}{r_2}\).
• Subtract the arguments: The argument of \(\frac{z_1}{z_2}\) is \(\theta_1 - \theta_2\). Adjust it if needed to fall within the required range of angles.

The resulting polar form of the quotient will be \(\frac{r_1}{r_2}(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))\), which elegantly compactly shows the effect of division on the complex numbers.

Magnitude and Argument

Understanding the magnitude and argument of a complex number is essential. These concepts provide essential insights into the geometric interpretation of complex numbers.The magnitude of a complex number, often denoted as \(r\), represents its size or length when visualized in the complex plane. For a complex number \(z = a + bi\), the magnitude is computed using the formula:\[r = \sqrt{a^2 + b^2}\]The argument of a complex number, denoted as \(\theta\), is the angle formed with the positive real axis. Its calculation requires:

• Using trigonometric functions like \(\tan^{-1}\left(\frac{b}{a}\right)\) to find the angle.
• Adjusting for the correct quadrant based on the signs of \(a\) and \(b\).

These two parts help convert a standard complex number into its polar form, aiding in operations like finding quotients and simplifying calculations.

Argument of Complex Numbers

The argument of a complex number is one of its defining attributes. It is the angle that the vector representing the complex number makes with the positive real axis in the complex plane.To correctly find the argument, consider:

• The function \(\tan^{-1}\left(\frac{b}{a}\right)\) gives the basic angle.
• The actual quadrant of the complex number, since this might affect the angle calculation:
  • Quadrant I: the angle is positive.
  • Quadrant II: add \(180^\circ\) to the basic angle.
  • Quadrant III: still add \(180^\circ\).
  • Quadrant IV: subtract the basic angle from \(360^\circ\).

Once calculated, the argument can help determine the number's behavior during mathematical operations, making it a vital part of understanding and working with complex numbers in polar form.