Problem 63 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$-5\left[\cos \left(\frac{4 \pi}{9}\right)+i \sin \left(\frac{4 \pi}{9}\right)\right]$$

Step-by-Step Solution

Answer

The rectangular form is approximately \(0.868 - 4.924i\).

Step 1: Understand the Problem

The complex number is given in polar form as \(-5[\cos(\frac{4\pi}{9}) + i\sin(\frac{4\pi}{9})]\). We need to convert it to rectangular form, which has the format \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

Step 2: Identify the Form of the Complex Number

Each complex number in polar form \(r(\cos(\theta) + i\sin(\theta))\) can be converted to rectangular form as \(r\cos(\theta) + ir\sin(\theta)\). In our case, \(r = -5\) and \(\theta = \frac{4\pi}{9}\).

Step 3: Use the Calculator to Find \(r\cos(\theta)\)

Calculate \(-5\cos(\frac{4\pi}{9})\) using a calculator. Ensure your calculator is in radian mode. The value is approximately \(-5 \times (-0.1736) = 0.868\).

Step 4: Use the Calculator to Find \(r\sin(\theta)\)

Calculate \(-5\sin(\frac{4\pi}{9})\) similarly using the calculator. The value is approximately \(-5 \times 0.9848 = -4.924\).

Step 5: Construct the Rectangular Form

Combine the results from Steps 3 and 4 to form the rectangular expression: \(0.868 - 4.924i\). Here, \(0.868\) is the real part and \(-4.924i\) is the imaginary part.

Key Concepts

Polar Form

When working with complex numbers, the polar form is a very useful representation. It expresses the number in terms of a magnitude (or length) and an angle.
In this form, a complex number is given as \( r (\cos(\theta) + i \sin(\theta)) \), where:

• \( r \) is the magnitude or modulus of the complex number.
• \( \theta \) is the angle, also known as the argument, usually measured in radians.

Polar form is particularly advantageous for multiplication and division of complex numbers, as well as for finding powers and roots. Using Euler's formula, the polar form can also be expressed as \( re^{i\theta} \). This notation simplifies many calculations in mathematics and physics.

Rectangular Form

The rectangular form of a complex number is perhaps the most intuitive way to represent it, using the format \( a + bi \). In this expression:

• \( a \) is the real part of the complex number.
• \( bi \) is the imaginary part, with \( i \) representing the imaginary unit where \( i^2 = -1 \).

To convert from polar to rectangular form, you can use the formulas \( a = r \cos(\theta) \) and \( b = r \sin(\theta) \).
This representation is particularly helpful for straightforward addition and subtraction of complex numbers. It's also useful in calculus when dealing with complex functions, as it aligns closely with the Cartesian coordinate system.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental in converting complex numbers from polar to rectangular form.
The cosine function helps calculate the real part of the number, while the sine function determines the imaginary part. Understanding these functions is essential for working with angles in radians:

• \( \cos(\theta) \) gives the adjacent side over the hypotenuse in a right triangle.
• \( \sin(\theta) \) gives the opposite side over the hypotenuse.

In the world of complex numbers, these functions help map the magnitude and direction outlined in polar form to a position on the complex plane, making conversion to rectangular form possible.

Calculus

In calculus, complex numbers often feature in the study of functions of a complex variable. The operations of differentiation and integration can extend naturally to the complex domain.
When dealing with complex numbers in polar or rectangular form, calculus techniques can be applied to analyze the behavior and properties of complex functions, including their continuity and differentiability.

• In polar form, expressions can simplify the process of integrating complex functions involving trigonometric expressions.
• Rectangular form provides a straightforward path to applying the rules of calculus by treating \( a + bi \) as a sum of two components, facilitating operations like differentiation and integration.

This synergy of concepts allows for effective problem-solving in advanced mathematical fields like signals processing and control theory.