Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Write each complex number in rectangular form. Give exact values for the real and imaginary parts. Do not use a calculator. $$\sqrt{3} \operatorname{cis} \frac{3 \pi}{2}$$

Step-by-Step Solution

Answer

The rectangular form is \( -\sqrt{3}i \).

Step 1: Understanding Polar to Rectangular Conversion

To convert a complex number from polar form to rectangular form, we use the formula: \[ z = r(\cos \theta + i \sin \theta) \] where \( r \) is the modulus and \( \theta \) is the argument.

Step 2: Identifying Components

In the given complex number \( \sqrt{3} \operatorname{cis} \frac{3 \pi}{2} \),\( r = \sqrt{3} \) and \( \theta = \frac{3\pi}{2} \).

Step 3: Calculating Real Part

The real part is given by \( r \cos \theta \). Substitute \( r = \sqrt{3} \) and \( \theta = \frac{3\pi}{2} \):\[ \text{Real Part} = \sqrt{3} \cos \frac{3 \pi}{2} = \sqrt{3} \times 0 = 0 \]Note: \( \cos \frac{3 \pi}{2} = 0 \).

Step 4: Calculating Imaginary Part

The imaginary part is given by \( r \sin \theta \). Substitute \( r = \sqrt{3} \) and \( \theta = \frac{3\pi}{2} \):\[ \text{Imaginary Part} = \sqrt{3} \sin \frac{3 \pi}{2} = \sqrt{3} \times (-1) = -\sqrt{3} \]Note: \( \sin \frac{3 \pi}{2} = -1 \).

Step 5: Expressing in Rectangular Form

Combine the real and imaginary parts to write the complex number in rectangular form:\[ z = 0 - \sqrt{3}i \]Hence, the rectangular form of the given complex number is \( -\sqrt{3}i \).

Key Concepts

Polar to Rectangular Conversion

When dealing with complex numbers, converting from polar to rectangular form is a fundamental skill. In polar form, a complex number is expressed as \( r \operatorname{cis} \theta \), where \( r \) represents the magnitude or modulus of the complex number, and \( \theta \) is the argument or angle in radians.

To convert from polar to rectangular form, we use the formula:

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

This formula means that:

• The real part of the complex number is \( r \cos \theta \).
• The imaginary part of the complex number is \( r \sin \theta \).

Understanding these basics ensures that you can interpret and transform complex numbers between different forms easily. It's essentially about breaking down a number into its components based on its magnitude and direction.

Real and Imaginary Parts

In any complex number, the real and imaginary parts play crucial roles. Understanding how to identify these parts in both rectangular and polar forms is vital for mathematical fluency.

The real part of a complex number in rectangular form, \( a + bi \), is \( a \), while the imaginary part is \( b \).

To pinpoint these in polar form involves some simple trigonometry:

• For the real part, calculate \( r \cos \theta \).
• For the imaginary part, calculate \( r \sin \theta \).

Through these two components, any polar form can be expressed equivalently as a rectangular form. For instance, in the example given, replacing \( \theta \) and \( r \) with their respective values helps us derive a clear rectangular form.

Trigonometric Functions

Trigonometric functions are pivotal in converting complex numbers from polar to rectangular forms. The sine and cosine functions help us understand how the complex number is oriented in the plane.

In any conversion:

• \( \cos \theta \) helps determine the "horizontal" component of the complex number, or the real part.
• \( \sin \theta \) provides the "vertical" component, or the imaginary part.

With the given exercise, \( \theta = \frac{3\pi}{2} \), an angle where:

• \( \cos \frac{3\pi}{2} = 0 \)
• \( \sin \frac{3\pi}{2} = -1 \)

This leads to a completely imaginary rectangular result. Understanding these functions and their values at key angles simplifies the conversion process significantly.