Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and leave the result in trigonometric form. $$\left[\frac{2}{3}\left(\cos \frac{6 \pi}{7}+i \sin \frac{6 \pi}{7}\right)\right]\left[9\left(\cos \frac{9 \pi}{14}+i \sin \frac{9 \pi}{14}\right)\right]$$

Step-by-Step Solution

Answer

The result of the operation is \(6(\cos \frac{147\pi}{98} + i \sin \frac{147\pi}{98})\).

Step 1: Understand the Trigonometric Form of Complex Numbers

A complex number can be expressed in trigonometric form as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude of the complex number and \(\theta\) is the angle it makes with the real axis. Here, the two complex numbers are given as \(\frac{2}{3}(\cos \frac{6 \pi}{7}+i \sin \frac{6 \pi}{7})\) and \(9(\cos \frac{9 \pi}{14}+i \sin \frac{9 \pi}{14})\).

Step 2: Multiply the Magnitudes and Add the Angles

According to De Moivre's theorem, when multiplying complex numbers in polar or trigonometric form, you simply multiply the magnitudes and add the angles. Hence, the magnitude of the result would be \(\frac{2}{3}\times9=6\). As for the angle, it would be \(\frac{6 \pi}{7}+\frac{9 \pi}{14} = \frac{84\pi+63\pi}{98}=\frac{147\pi}{98}\).

Step 3: Write the Result

Finally, the result is written in the form \( r(\cos \theta + i \sin \theta) \). This completes the multiplication operation on the two complex numbers in trigonometric form.

Key Concepts

De Moivre's Theorem

De Moivre's theorem is a powerful formula used to compute powers and roots of complex numbers when they are in polar or trigonometric form. It states that for a complex number expressed as \(r(\cos \theta + i \sin \theta)\), its nth power can be found by raising the magnitude \(r\) to the power of \(n\) and multiplying the angle \(\theta\) by \(n\). Expressly, \( (r(\cos \theta + i \sin \theta))^n = r^n (\cos(n\theta) + i \sin(n\theta)) \).

This theorem is not just limited to integer powers; it can be adapted to work for roots by using fractional powers. The elegance of De Moivre's theorem lies in its simplicity which translates to computational ease, avoiding the more cumbersome algebraic manipulation that may come with using standard form complex numbers. When De Moivre's theorem is applied to the multiplication of two complex numbers, it simplifies the process by allowing us to multiply the magnitudes and add the angles directly, streamlining the operation significantly.

Multiplying Complex Numbers

When working with complex numbers, one may encounter various operations such as addition, subtraction, multiplication, and division. Particularly for multiplication, the process is greatly simplified in the polar or trigonometric form. Rather than expanding binomials and combining like terms as with the standard form \(a+bi\), we apply a rule that derives from De Moivre's theorem.

Simple Rule for Multiplication:

• Multiply the magnitudes (or moduli) of the complex numbers.
• Add the angles (or arguments) of the complex numbers.

This method echoes the geometric interpretation of complex numbers: magnitude corresponds to the distance from the origin on the complex plane, and the angle represents the direction. As seen in the exercise provided, the resultant complex number after multiplication can be swiftly written as a single complex number in trigonometric form, \( r(\cos \theta + i \sin \theta) \).

Polar Form of Complex Numbers

The polar form of complex numbers offers an alternative representation to the familiar rectangular form \(a+bi\). Instead of x and y coordinates, we describe a complex number in terms of its distance from the origin (the magnitude or modulus) and its angle with the positive real axis (the argument).

This polar form is written as \( r(\cos \theta + i \sin \theta) \) where \( r = \sqrt{a^2+b^2} \) and \( \theta = \arctan\left(\frac{b}{a}\right) \) for a complex number \(a+bi\). The beauty of this representation becomes even more apparent when dealing with multiplication of complex numbers, as it aligns with the intuitive understanding of rotating and scaling vectors on the complex plane. It also sets the stage for using De Moivre's theorem, which relies on this polar form to simplify complex number calculations such as those involving powers and roots.