Problem 55 - Precalculus with Limits

Question

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{\cos\ \pi + i\ \sin\ \pi}{\cos(\pi/3) + i\ \sin(\pi/3)}\)

Step-by-Step Solution

Answer

The result of the operation is \(\cos(2\pi/3) + i\ \sin(2\pi/3)\)

Step 1: Calculate modulus and argument for each complex number

For the number \(\cos\ \pi + i\ \sin\ \pi\), the modulus is 1 (since it's a unit circle) and the argument is \(\pi\) (from the given sine and cosine values). Similarly, for the number \(\cos(\pi/3) + i\ \sin(\pi/3)\), the modulus is also 1 and the argument is \(\pi/3\).

Step 2: Perform the Division

The division of two complex numbers in polar form is performed by dividing the moduli and subtracting the arguments. Here, the modulus of the result is \(1/1 = 1\), and the argument is \(\pi - \pi/3 = 2\pi/3\).

Step 3: Convert the Result into Trigonometric Form

The resulting complex number in trigonometric form is \(1 * (\cos(2\pi/3) + i\ \sin(2\pi/3))\).

Key Concepts

Polar Coordinates

Polar coordinates provide a way to represent complex numbers and points in a two-dimensional plane. Instead of using the traditional Cartesian coordinates

• where a point is defined by its horizontal (x) and vertical (y) distances,

polar coordinates use a distance from the origin (called the radius or modulus) and an angle from a reference direction (called the argument or angle).
This system is quite useful when dealing with rotations and periodic phenomena, especially in physics and engineering.

• The radius, often denoted as \(r\), determines how far a point is from the origin.
• The argument, usually represented by \(\theta\), describes the angle made with the positive x-axis.

In the context of complex numbers, converting to polar coordinates involves identifying these two components to simplify mathematical operations like multiplication and division.

Complex Numbers

Complex numbers extend the idea of real numbers by adding an imaginary part. They take the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
The imaginary unit \(i\) is defined by the property \(i^2 = -1\). This concept allows for the solution of equations that do not have real solutions, such as roots of negative numbers.

• Complex numbers can also be represented in trigonometric or polar form, using their modulus and argument.
• This polar form makes calculating operations such as multiplication, division, and roots more straightforward by manipulating angles and moduli rather than lengthy algebraic expressions.

In this form, any complex number can be expressed as \(r (\cos \theta + i \sin \theta)\), where \(r\) is its magnitude or distance from the origin, and \(\theta\) is its angle from the positive x-axis.

Trigonometric Functions

Trigonometric functions, primarily sine and cosine, are fundamental in mathematics and are closely related to circular and wave phenomena. They describe ratios in right triangles, with values ranging between -1 and 1.
These functions have periodic behavior, repeating their values systematically as they trace the circle's circumference.

• When combined with complex numbers, sine and cosine help express numbers in the unit circle, a circle with a radius of one centered on the origin.
• This connection is evident in the Euler's formula, \(e^{i\theta} = \cos\theta + i\sin\theta\), linking exponential functions to trigonometry.

Trigonometric functions also facilitate conversion between rectangular and polar forms, allowing easier manipulation, especially in wave and oscillation problems, showing their wide-ranging applicability in science and engineering.