Problem 48 - Precalculus with Limits

Question

In Exercises \(47-58\) , perform the operation and leave the result in trigonometric form. $$ \left[\frac{3}{4}\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\right]\left[4\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right] $$

Step-by-Step Solution

Answer

The result of the operation is \(3(\cos\ \frac{7\pi}{12} + i\ \sin\ \frac{7\pi}{12})\).

Step 1: Identify the Moduli and Arguments

The moduli of the given complex numbers are \(\frac{3}{4}\) and \(4\), and the arguments are \(\frac{\pi}{3}\) and \(\frac{3\pi}{4}\) respectively.

Step 2: Multiply the Moduli and Add the Arguments

Multiply the moduli \(\frac{3}{4} \times 4 = 3\). Add the arguments \(\frac{\pi}{3} + \frac{3\pi}{4} = \frac{7\pi}{12}\). The resulting complex number in trigonometric form is then \(3(\cos\frac{7\pi}{12} + i\sin\frac{7\pi}{12})\).

Step 3: Simplify

You have now completed all the calculations, and found the product of the two complex numbers in trigonometric form. The expression does not require any further simplification.

Key Concepts

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number offers a more intuitive understanding of its geometric representation. Instead of the usual 'a + bi' format, a complex number can be expressed as \( r(\text{cos}\theta + i\text{sin}\theta) \), where \( r \) is the modulus and \( \theta \) the argument of the complex number.

This form is particularly useful when dealing with multiplication and division, as the rules of these operations are greatly simplified. For instance, when multiplying complex numbers in trigonometric form, you simply multiply the moduli and add the arguments, a concept that will be further detailed in one of the next sections.

Multiplying Complex Numbers

To multiply complex numbers in their trigonometric forms, a straightforward process is used: multiply the moduli and add the arguments. This method is based on the properties of exponents and the Euler's formula. For example, given two complex numbers \( z_1 = r_1(\text{cos} \theta_1 + i\text{sin} \theta_1) \) and \( z_2 = r_2(\text{cos} \theta_2 + i\text{sin} \theta_2) \), the product is \( z_1 \times z_2 = r_1r_2(\text{cos}(\theta_1+\theta_2) + i\text{sin}(\theta_1+\theta_2)) \).

This operation can be visualized geometrically as rotating and scaling vectors in the complex plane. Multiplication by a complex number can be thought of as scaling the length of a vector by the modulus, while simultaneously rotating the vector by an angle equal to the argument.

Modulus of Complex Number

The modulus of a complex number, denoted as \( |z| \), is the distance from the origin to the point representing the complex number in the complex plane. It is equivalent to the hypotenuse of a right triangle whose legs are the real and imaginary parts of the complex number. Mathematically, if \( z = a + bi \), its modulus is \( |z| = \sqrt{a^2 + b^2} \).

In trigonometric form, the modulus represents the radius of the circle in the polar coordinate system, and it simplifies calculations involving multiplication, as seen in the given exercise. When multiplying complex numbers, one begins by multiplying their moduli, which can also be seen as combining their respective scalings.

Argument of Complex Number

The argument of a complex number is the angle formed by the real axis and the vector representing the complex number in the complex plane. It is usually denoted by \( \theta \) and is measured in radians. The argument can be visualized as the direction of the vector from the origin to the complex number.

It is crucial to note that, while the modulus of a complex number is always non-negative, the argument can take any value between \( -\pi \) and \( \pi \) for a principal argument, or can be expressed modulo \( 2\pi \) to include all possible angles. When multiplying complex numbers, adding their arguments effectively achieves the resultant direction after the multiplication process.