Problem 53 - Precalculus with Limits

Question

Multiplying or Dividing Complex Numbers Exercises \(45-56,\) perform the operation and leave the result in trigonometric form. $$\frac{\cos \pi+i \sin \pi}{\cos (\pi / 3)+i \sin (\pi / 3)}$$

Step-by-Step Solution

Answer

The result of the division is \(cos(2\pi / 3) + isin(2\pi / 3)\).

Step 1: Identifying the components of the complex numbers

The radians in the cosine and sine functions represent the angles for the complex numbers. Therefore, for the complex number in the numerator, \(a_1 = \pi\) and for the complex number in the denominator, \(a_2 = \pi / 3\). The magnitudes for both complex numbers are equal to 1 as there are no multipliers.

Step 2: Applying the formula for division of complex numbers

We insert the identified values into the formula for the division of complex numbers. This gives us \(\frac{1 * (cos(\pi) + isin(\pi))}{1 * (cos(\pi / 3) + isin(\pi / 3))} = 1 * (cos(\pi - \pi / 3) + isin(\pi - \pi / 3))\).

Step 3: Simplifying the expression

By making the subtraction in the cosine and sine functions, we obtain \(cos(2\pi / 3) + isin(2\pi / 3)\).

Key Concepts

Complex Numbers

Complex numbers are a cornerstone of higher mathematics, particularly useful in fields like engineering, physics, and computer science. They consist of two parts: a real part and an imaginary part. Formally, a complex number can be expressed as a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with the property that i^2 = -1.

Understanding complex numbers enhances the ability to solve equations that have no real solutions. They come in handy especially when dealing with oscillations and waves, electronic currents, quantum mechanics, and many other scientific and mathematical scenarios.

Trigonometric Form

When we express complex numbers in trigonometric form, we showcase them in terms of their magnitude (also called modulus) and angle (also referred to as the argument). A complex number in trigonometric form looks like this: r(cos(θ) + i sin(θ)), where r represents the magnitude of the complex number, and θ represents the angle it forms with the positive x-axis on the complex plane.

The trigonometric form is particularly useful for multiplying, dividing, and raising complex numbers to powers because it simplifies the actual arithmetic involved. It also provides an intuitive geometric interpretation of complex number operations.

Cosine and Sine Functions

The cosine and sine functions are fundamental to trigonometry, describing the relationships among the sides of a right triangle to its angles. They're also essential in defining the position of points on the unit circle. For any angle θ, the sine function, denoted as sin(θ), gives the vertical coordinate of the point on the unit circle, while the cosine function, denoted as cos(θ), gives the horizontal coordinate.

Their importance extends far beyond triangles; these functions describe wave phenomena, oscillations, circular motion, and much more across various scientific disciplines.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which are based on dividing a circle into 360 equal parts, radians provide a direct measure of the distance along the circle's circumference. One radian is the angle created when the arc length is equal to the radius of the circle.

Radians are the preferred unit of angular measurement in mathematics because they simplify many formulae and calculations. In particular, they make the understanding and calculation of trigonometric functions much more intuitive, as these functions have simple and elegant representations in terms of radians.