Problem 27 - Algebra and Trigonometry

Question

In Exercises \(27-36,\) write each complex number in rectangular form. If necessary, round to the nearest tenth. $$ 6\left(\cos 30^{\circ}+i \sin 30^{\circ}\right) $$

Step-by-Step Solution

Answer

The rectangular form of the given complex number is \(a + ib\), where \(a = 6 \cos 30^\circ\) and \(b = 6 \sin 30^\circ\).

Step 1: Calculate Real Part

The real part \(a\) of the rectangular form of the complex number is calculated by \(a = r \cos \theta\). Here, \(r = 6\) and \(\theta = 30^\circ\). So, \(a = 6 \cos 30^\circ\).

Step 2: Calculate Imaginary Part

The imaginary part \(b\) of the rectangular form of the complex number is calculated by \(b = r \sin \theta\). Given \(r = 6\) and \(\theta = 30^\circ\), this means \(b = 6 \sin 30^\circ\).

Step 3: Write in Rectangular Form

Now we express the rectangular form of complex number using the real part from Step 1 and imaginary part from Step 2. The complex number in rectangular form is \(a + ib\).

Key Concepts

Complex Numbers

Complex numbers are the building blocks of a mathematical area that extends the traditional number system beyond real numbers. Imagine trying to solve an equation like \(x^2 + 1 = 0\); in the realm of real numbers, there's no solution, because no real number squared can give a negative result. But in the world of complex numbers, there is an answer: \(x = i\), where \(i\) is the imaginary unit, defined as \(i^2 = -1\).

A complex number is typically written as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. Each part on its own is a real number, with the imaginary part being multiplied by the imaginary unit \(i\). This system allows for a much larger set of solutions to equations and becomes invaluable in fields like engineering, physics, and computer science.

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number is an elegant expression that uses the polar coordinates system. Instead of defining a point (or number) using the standard \(x\)- and \(y\)-coordinates (the rectangular form), it uses the angle \(\theta\)—known as the argument—and the distance \(r\) from the origin, which is the magnitude or modulus.

The complex number in trigonometric form is written as \(r(\text{cos } \theta + i\text{sin } \theta)\). This format shows the connection between complex numbers and geometry and simplifies the multiplication and division of complex numbers due to the properties of cosine and sine.

Converting Complex Numbers

Converting complex numbers from their trigonometric form to the more familiar rectangular form is a direct process involving basic trigonometric functions. You'll often perform this conversion to simplify the addition, subtraction, or comparison of complex numbers.

To make the conversion, you multiply the magnitude \(r\) by the cosine of the angle to find the real part, and by the sine of the angle to find the imaginary part. Mathematically, if you have a complex number in trigonometric form \(r(\text{cos } \theta + i\text{sin } \theta)\), you convert it by setting the real part \(a\) to \(r \cos \theta\) and the imaginary part \(b\) to \(r \sin \theta\), resulting in the rectangular form \(a + bi\).

This process is wonderfully demonstrated in the example where the complex number \(6(\text{cos } 30^{\circ} + i\text{sin } 30^{\circ})\) is converted into rectangular form. Here, the magnitude is 6, and the argument is 30 degrees, leading to the rectangular components after applying the respective trigonometric functions.

Imaginary and Real Parts

Every complex number can be deconstructed into two main components: the real part and the imaginary part. The real part is simply a traditional real number and can be represented on the horizontal axis of the complex plane. The imaginary part, however, is a real number that has been multiplied by the imaginary unit \(i\), located on the vertical axis if we're visualizing it on the complex plane.

The interplay between these parts dictates the position and representation of a complex number. When you work with them in algebraic operations, they're treated almost like separate entities, but they come together to form the unique entity of a complex number. Understanding how to manipulate and interpret these parts is crucial for working with complex numbers in any context, whether it be mathematical, scientific, or engineering problems.