Problem 52 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right) $$

Step-by-Step Solution

Answer

The expression is equivalent to \(0 - 3i\).

Step 1: Understand the Expression

We need to express the given complex number in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The expression provided is in polar form: \(3(\cos \frac{3 \pi}{2} + i \sin \frac{3 \pi}{2})\).

Step 2: Evaluate the Trigonometric Functions

Find the values of \(\cos \frac{3 \pi}{2}\) and \(\sin \frac{3 \pi}{2}\). Taking \(\cos \frac{3 \pi}{2}\) and \(\sin \frac{3 \pi}{2}\), we get \(\cos \frac{3 \pi}{2} = 0\) and \(\sin \frac{3 \pi}{2} = -1\).

Step 3: Substitute the Values

Substitute the evaluated trigonometric function values back into the expression: \(3(0 + i(-1))\). This simplifies to \(3(0 - i)\).

Step 4: Simplify the Expression

Distribute the 3 across the terms inside the parenthesis: \(3 \times 0 + 3 \times (-i) = 0 - 3i\).

Step 5: Final Form

The expression in the form \(a + bi\) is \(0 - 3i\), which is \(0 - 3i\). Thus, \(a = 0\) and \(b = -3\).

Key Concepts

Polar Form

Polar form is a way to represent complex numbers as a product of a magnitude and an angle. It is written as \( r(\cos \theta + i \sin \theta) \). This can be a more intuitive form when dealing with multiplications and divisions of complex numbers.

• Magnitude \( (r) \): Represents the distance of the complex number from the origin in the complex plane.
• Angle \( (\theta) \): Also known as the argument, it is the angle formed with the positive real axis.

Instead of x and y coordinates, polar form uses \( r \) and \( \theta \) to describe the point's location.
In the exercise, the complex number is given in polar form: \( 3(\cos \frac{3 \pi}{2} + i \sin \frac{3 \pi}{2}) \). Here, \( r = 3 \) and \( \theta = \frac{3 \pi}{2} \).

Trigonometric Functions

Trigonometric functions like cosine and sine are fundamental in polar form to express the imaginary and real components. Particularly, these functions help in defining angles and using them to compute specific points on the unit circle at those angles.

• \( \cos \theta \): Determines the horizontal component on the unit circle.
• \( \sin \theta \): Determines the vertical component on the unit circle.

For the angle \( \frac{3 \pi}{2} \), in the unit circle:

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

These values are crucial for simplifying polar equations into the rectangular form.

Conversion to Rectangular Form

Conversion to rectangular form involves taking a complex number in polar form and expressing it in terms of real and imaginary parts. This is typically done using the trigonometric values of cosine and sine.
To convert: 1. Multiply the magnitude, \( r \), by \( \cos \theta \) to find the real part.2. Multiply the magnitude, \( r \), by \( i \sin \theta \) to find the imaginary part.
In this example, after evaluating the trigonometric functions:\[ 3 \left( 0 + i(-1) \right) = 3 \times 0 + 3 \times (-i) = 0 - 3i \]Thus, the rectangular form is \( 0 - 3i \), with\( a = 0 \),\( b = -3 \).

Imaginary Numbers

Imaginary numbers are a key component in complex numbers, represented by the imaginary unit \( i \), where \( i = \sqrt{-1} \). An imaginary number is any real number multiplied by \( i \).
They allow us to extend the real number system \( \mathbb{R} \) to the complex number system, \( \mathbb{C} \).

• \( 0i \) indicates a purely real number.
• \( ai \) where \( a eq 0 \) shows the imaginary part of a complex number.

In our solution, the imaginary component is clear: the number is any multiple of \( i \). Here, \( -3i \) emphasizes the imaginary number \( -3i \).
It's essential to understand that without imaginary numbers, expressing certain types of equations would be impossible.