Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 6\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

The expression is \(-3 + 3\sqrt{3}i\).

Step 1: Recall Trigonometric Values

First, recall the trigonometric values for the angle \( \frac{2\pi}{3} \). Based on the unit circle, we have: \( \cos\frac{2\pi}{3} = -\frac{1}{2} \) and \( \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2} \).

Step 2: Substitute the Trigonometric Values

Next, substitute these values back into the expression: \[ 6\left(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\right) = 6\left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right). \]

Step 3: Distribute the Multiplication

Now, distribute the 6 across the terms inside the parenthesis:\[ 6\left(-\frac{1}{2}\right) + 6\left(i \frac{\sqrt{3}}{2}\right). \]

Step 4: Simplify the Expression

Simplify the expression by performing the multiplication:\[ -3 + i3\sqrt{3}. \]

Step 5: Express in the Form \(a+bi\)

Finally, combine the results into the standard form \(a+bi\): The expression becomes \[-3 + 3\sqrt{3}i.\]

Key Concepts

Trigonometric Form of Complex Numbers

Complex numbers can be expressed in different forms, one of which is the trigonometric form. This form is useful in mathematics for simplifying multiplication and division of complex numbers. It uses the polar coordinates of a complex number on the complex plane. Typically, this form is written as

• \( z = r(\cos\theta + i\sin\theta) \),

where \( r \) represents the magnitude (or modulus) of the complex number and \( \theta \) is the argument, or the angle formed with the positive real axis.
This representation is particularly advantageous when dealing with powers and roots of complex numbers, as it leverages Euler's formula. This allows us to multiply and divide complex numbers more efficiently by working with their angles and magnitudes rather than their rectangular coordinates directly.

Unit Circle

The unit circle is a fundamental concept in trigonometry and complex number theory. It is a circle with a radius of 1, centered at the origin of the coordinate plane (0,0). The unit circle makes it easy to define the trigonometric functions for all angles. It is particularly useful for understanding the geometric interpretation of complex numbers.
In the unit circle, each point on the circle corresponds to a complex number whose modulus is 1. The angle \( \theta \), typically in radians, describes the position of this point on the circle. For example,

• for \( \theta = \frac{2\pi}{3} \), the coordinate values based on the unit circle are \( \cos\frac{2\pi}{3} = -\frac{1}{2} \) and \( \sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2} \).

These values illustrate the beauty and simplicity of the unit circle, allowing us to easily find coordinates for complex numbers expressed in trigonometric forms.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved. They play a crucial role in simplifying complex expressions and solving trigonometric equations.
Some well-known identities include Pythagorean identities, angle sum and difference identities, and double and half-angle identities. These identities are essential in complex number arithmetic when expressed in trigonometric or polar form. For instance, recognizing that the identity

• \( \sin^2\theta + \cos^2\theta = 1 \)

helps verify unit circle values and apply transformations to solve problems involving trigonometric forms of complex numbers efficiently.
Understanding these identities allows students to deconstruct complex problems into manageable parts and aids in manipulating formulas to achieve the desired outcomes, such as converting expressions into the standard form \( a + bi \).