Problem 60 - Contemporary Precalculus

Question

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\frac{\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}}{\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}}$$

Step-by-Step Solution

Answer

The quotient of the division of the given complex numbers in polar form is \(1(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\). In rectangular form, the quotient is \(0 + 1i\).

Step 1: Find the Magnitude and Argument of the Quotient

Given the fraction: $$\frac{(\cos \frac{3\pi}{4} + i\sin \frac{3\pi}{4})}{(\cos \frac{\pi}{4} + i\sin \frac{\pi}{4})}$$ The magnitude and argument of both numerator and denominator are 1 and their respective arguments. Now, we need to divide the two complex numbers in polar form. To divide two complex numbers in polar form, we divide their magnitudes and subtract their arguments: Magnitude: \(\frac{r_1}{r_2} = \frac{1}{1} = 1\) Argument: \(\theta_1 - \theta_2 = \frac{3\pi}{4} - \frac{\pi}{4} = \frac{2\pi}{4} = \frac{\pi}{2}\) The quotient in polar form is \(1(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\).

Step 2: Convert the Quotient to Rectangular Form

To convert the quotient from polar form to rectangular form, we use the formulas: $$a = r\cos \theta$$ $$b = r\sin \theta$$ In our case, \(r = 1\) and \(\theta = \frac{\pi}{2}\). Plugging these values into the formulas, we get: $$a = 1\cos \frac{\pi}{2} = 0$$ $$b = 1\sin \frac{\pi}{2} = 1$$ The quotient in rectangular form is \(0 + 1i\). So, Polar Form: \(1(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2})\) Rectangular Form: \(0 + 1 i\)

Key Concepts

Polar Form

The polar form of a complex number is a unique representation that highlights the relationship between the magnitude and the argument of the number. In essence, any complex number in polar form is expressed as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the argument.

• Magnitude: This is the absolute value or length of the complex number.
• Argument: The angle formed by the number with the positive x-axis.

Using polar form allows for straightforward multiplication and division of complex numbers, relying only on multiplying/dividing magnitudes and adding/subtracting arguments.
In our exercise, the complex numbers are already given in polar form, making division simple and neat. The result is another complex number expressed in polar form.

Rectangular Form

The rectangular form of a complex number is how complex numbers are typically represented, especially in basic algebra. It is characterized by \(a + bi\), where:

• \(a\) is the real part of the number.
• \(b\) is the imaginary part.

In rectangular form, complex numbers appear on a Cartesian coordinate system, with axes for real and imaginary parts.
To transform a complex number from polar to rectangular form, you calculate the real part as \(r\cos \theta\) and the imaginary part as \(r\sin \theta\).
In the solution provided for the exercise, this conversion process yielded \(0 + 1i\), which means the complex number is purely imaginary, reaching one unit along the imaginary axis.

Complex Division

Complex division, which involves dividing one complex number by another, can be quite intricate in their rectangular form. However, the polar form significantly simplifies it. The rule is to divide the magnitudes (\(r\)) and subtract the arguments (\(\theta\)).
This approach gives us a straightforward result:

• Magnitude: \(\frac{r_1}{r_2}\)
• Argument: \(\theta_1 - \theta_2\)

As seen in this exercise, both numerator and denominator had magnitudes of 1. Their arguments were \(\frac{3\pi}{4}\) and \(\frac{\pi}{4}\), respectively. Subtracting them resulted in the final argument of \(\frac{\pi}{2}\). Therefore, the division leads to a result of \(1(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\).

Magnitude and Argument

Magnitude and argument are pivotal concepts in understanding complex numbers, especially in polar coordinates. The magnitude, often denoted by \(|z|\) or \(r\), is the distance from the origin to the point representing the complex number in the complex plane.
It is computed as \(\sqrt{a^2 + b^2}\) when expressed in rectangular form as \(a + bi\).

The argument of a complex number, commonly denoted as \(\theta\), represents the angle from the positive real axis to the line segment that joins the origin with the point. It's measured in radians.
In polar coordinates, this angle gives an insight into the direction of the complex number. For the exercise, \(\frac{3\pi}{4}\) and \(\frac{\pi}{4}\) were the arguments for the two complex numbers involved, which upon subtraction gave \(\frac{\pi}{2}\).
Remembering these basic characteristics facilitates operations on complex numbers, especially when converting between forms.