Problem 34 - Contemporary Precalculus

Question

In Exercises \(25-36,\) express the number in the form \(a+b i\). $$3(\cos 5+i \sin 5)$$

Step-by-Step Solution

Answer

Question: Express the complex number in polar form: \(3(\cos 5 + i\sin 5)\) in the form \(a + bi\). Answer: \(3\cos(5) + 3i\sin(5)\)

Step 1: Identify the magnitude and argument

The given complex number in polar form is \(3(\cos 5 + i\sin 5)\). The magnitude is \(3\) and the argument is \(5\).

Step 2: Apply the cosine and sine functions to the argument

To find the real and imaginary parts, apply the cosine and sine functions to the argument and multiply by the magnitude: Real part (a): \(3\cos(5)\) Imaginary part (b): \(3\sin(5)\)

Step 3: Write the complex number in the form \(a+bi\)

Now, we have found the real and imaginary parts, so we can express the complex number in the form \(a+bi\). The complex number is: \(3\cos(5) + 3i\sin(5)\)

Key Concepts

Understanding Complex Numbers

A complex number is a mathematical construct that expands on the idea of quantity to include numbers that are not real. This may seem abstract at first, but complex numbers are a natural extension of the real numbers we use every day. In essence, a complex number is composed of two parts: a real part (a) and an imaginary part (b), where the imaginary unit is represented by the symbol 'i', which is defined as the square root of -1. The standard format of a complex number is written as \(a + bi\), where 'a' stands for the real component and 'b' represents the coefficient of the imaginary part.

The beauty of complex numbers lies in their ability to represent two-dimensional numbers, combining the concept of magnitude and direction. They have fascinating properties and are used in various areas of science and engineering, such as electrical engineering, quantum physics, and applied mathematics. Understanding complex numbers thus opens a door to advanced scientific fields and complex problem-solving.

Diving into Polar Coordinates

Polar coordinates offer a different perspective for representing complex numbers, emphasizing their magnitude and the angle they form with the real axis, known as the argument. The polar form is particularly useful for multiplication, division, and powers of complex numbers.

In polar coordinates, any point on a plane is described by the radius (r), which measures the distance from the origin, and an angle (θ), which defines the direction from the origin. A complex number in polar form thus looks like \(r(cos θ + i sin θ)\). This is known as the polar representation of a complex number, with 'r' being analogous to the magnitude and 'θ' known as the argument, or the phase angle, which is usually measured in radians.

Converting between the rectangular form (a + bi) and the polar form (r(cos θ + i sin θ)) involves trigonometry, but makes operations involving complex numbers significantly more straightforward in many cases. In electrical engineering, for example, this form is instrumental for understanding waveforms and alternating current (AC) circuits.

Trigonometry and Complex Numbers

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. It is essential for understanding the polar form of complex numbers. In the context of complex numbers, trigonometry provides the tools to describe the location of a point on the complex plane through its angle (argument) and distance from the origin (magnitude).

The sine and cosine functions are particularly important, as they relate the angle to the rectangular coordinates: cosine corresponds to the 'x' coordinate (real part) and sine corresponds to the 'y' coordinate (imaginary part). Here is how we can translate a complex number from its polar form to its standard form using trigonometry: Given a complex number in polar form, \(r(cos θ + i sin θ)\), you take the magnitude 'r' and multiply it by the cosine of the angle to find the real part, and by the sine of the angle to find the imaginary part. This process of converting the representation of a complex number from polar to rectangular form (or vice versa) is a fundamental application of trigonometry in complex analysis.

Real-world Application

Trigonometry and complex numbers intersect in the real world where waves, vibrations, and oscillations are analyzed. For example, in electrical engineering, complex numbers and trigonometry are used to analyze AC circuits, where the voltage and current waveforms are out of phase. Understanding how these mathematical concepts work together not only strengthens problem-solving skills but also enables practical applications in technology and science.