Problem 41 - Precalculus

Question

Express each complex number in rectangular form. $$5\left(\cos 180^{\circ}+i \sin 180^{\circ}\right)$$

Step-by-Step Solution

Answer

The rectangular form is \(-5\).

Step 1: Convert to Rectangular Form

The given complex number is in trigonometric form: \( 5(\cos 180^{\circ} + i \sin 180^{\circ}) \). First, calculate \( \cos 180^{\circ} \) and \( \sin 180^{\circ} \). We know:\[\cos 180^{\circ} = -1 \quad \text{and} \quad \sin 180^{\circ} = 0.\]Substitute these values into the expression to get the rectangular form.

Step 2: Perform the Multiplication

Multiply the magnitude by the cosine and sine values to express the number in rectangular form: \[5 \times (-1 + i \times 0) = 5 \times (-1) + 5 \times 0i.\]Carry out the multiplication to simplify further.

Step 3: Simplify the Expression

Simplify the expression from the previous step: \[-5 + 0i.\] Therefore, the rectangular form of the given complex number is \(-5 + 0i\), which can be simply written as \(-5\).

Key Concepts

Rectangular Form

Complex numbers can be expressed in different forms, and the rectangular form is one of the most common. In rectangular form, a complex number is expressed as: \\( a + bi \)\, where:

• \( a \) represents the real part of the number.
• \( b \) represents the imaginary part of the number, with \( i \) being the imaginary unit that satisfies \( i^2 = -1 \).

This representation is called 'rectangular' because it can be visualized as a point on the complex plane, with \( a \) as the horizontal coordinate and \( b \) as the vertical coordinate. For instance, the complex number \(-5 + 0i\) can be simplified to \(-5\) as it lies entirely on the real axis. This simplification is helpful when performing arithmetic operations like addition, subtraction, multiplication, and division on complex numbers. Understanding the rectangular form of a complex number is crucial for solving many problems in mathematics, especially those involving complex plane analysis.

Trigonometric Form

Complex numbers can also be written in what is known as trigonometric or polar form. In this form, a complex number is expressed using a magnitude and an angle, represented as: \\( r(\cos \theta + i \sin \theta) \)\, where:

• \( r \) is the magnitude or modulus of the complex number, calculated as \( \sqrt{a^2 + b^2} \).
• \( \theta \) is the argument or angle, often measured in degrees or radians, noting the direction of the vector from the origin.

Using trigonometric identities, this form allows us to easily handle multiplication, division, and finding powers and roots of complex numbers. In our exercise, the trigonometric form \( 5(\cos 180^{\circ} + i \sin 180^{\circ}) \) indicated a complex number with a magnitude of 5 and an angle of 180 degrees, pointing entirely in the negative direction of the x-axis. Converting it to rectangular form involves computing the cosine and sine values at 180 degrees, which are \(-1\) and \(0\) respectively.

Multiplication of Complex Numbers

Multiplying complex numbers can be simplified using different forms. In the trigonometric form, multiplication becomes straightforward: multiply the magnitudes and add the angles. Given two complex numbers \\( r_1(\cos \theta_1 + i \sin \theta_1) \)\ and \\( r_2(\cos \theta_2 + i \sin \theta_2) \)\, their product is: \\( (r_1r_2)(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) \).For our exercise, applying this to \( 5(\cos 180^{\circ} + i \sin 180^{\circ}) \), we multiply the magnitude 5 by \(-1\) (the cosine of 180 degrees), resulting in \(-5 + 0i\). This simplifies easily to \(-5\), showing how trigonometric form can make complex number multiplication less cumbersome. Understanding these multiplication techniques is essential in fields like signal processing, quantum physics, and electrical engineering, where complex numbers frequently arise.