Problem 28 - Algebra and Trigonometry

Question

Write each complex number in rectangular form. $$ 2 e^{i \frac{5 \pi}{6}} $$

Step-by-Step Solution

Answer

-\sqrt{3} + i

Step 1: Identify the polar form components

The complex number is given in the polar form as \(2 e^{i \frac{5 \pi}{6}} \). Here, the magnitude (or modulus) is \(r = 2\) and the argument (or angle) is \(\theta = \frac{5 \pi}{6}\).

Step 2: Use Euler's formula

According to Euler's formula, \( e^{i \theta} = \cos \theta + i \sin \theta\). Substitute \(\theta = \frac{5 \pi}{6}\) into Euler's formula so that \( e^{i \frac{5 \pi}{6}} = \cos \left(\frac{5 \pi}{6}\right) + i \sin \left(\frac{5 \pi}{6}\right) \).

Step 3: Calculate the cosine component

Find \( \cos \left( \frac{5 \pi}{6} \right) \). Using the unit circle, \( \cos \left( \frac{5 \pi}{6} \right) = -\frac{\sqrt{3}}{2} \).

Step 4: Calculate the sine component

Find \( \sin \left( \frac{5 \pi}{6} \right) \). Using the unit circle, \( \sin \left( \frac{5 \pi}{6} \right) = \frac{1}{2} \).

Step 5: Combine magnitudes and components

Multiply the magnitude \(r = 2\) by the rectangular components \( 2 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right) \).

Step 6: Simplify to rectangular form

Distribute the 2 to get: \( 2 \left( -\frac{\sqrt{3}}{2} \right) + 2i \left( \frac{1}{2} \right) = -\sqrt{3} + i \). Thus, the rectangular form is \( -\sqrt{3} + i \).

Key Concepts

Polar form

When dealing with complex numbers, you will often see them in two main forms. One of these is the polar form. The polar form expresses a complex number as \( r e^{i \theta} \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the argument (or angle). Here is the step-by-step to understand and utilize it:
- The magnitude \( r \) represents the distance from the origin to the point in the complex plane.
- The angle \( \theta \) represents the direction of the line connecting the origin to the point.
In the exercise, the complex number is given as \( 2 e^{i \frac{5 \pi}{6}} \). This means the magnitude is 2 and the angle is \( \frac{5 \pi}{6} \).
The polar form allows for easy multiplication and division of complex numbers. However, to perform addition or subtraction, you will often need to convert the polar form into the rectangular form.

Euler's formula

Euler's formula is integral when working with complex numbers in polar form. The formula states: \( e^{i \theta} = \cos \theta + i \sin \theta \). This formula links exponential functions to trigonometric functions. For example, in our exercise, to convert the polar form \( 2 e^{i \frac{5 \pi}{6}} \) into rectangular form, we leverage Euler's formula:
- Substitute \( \theta = \frac{5 \pi}{6} \) so we have \( e^{i \frac{5 \pi}{6}} = \cos \left(\frac{5 \pi}{6}\right) + i \sin \left(\frac{5 \pi}{6}\right) \). This converts the exponential to trigonometric form, making it easier to work with.

Unit circle

The unit circle is a crucial concept in trigonometry and complex numbers. It's a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle allows us to find the sine and cosine of any angle. For the exercise:
- To find \( \cos \left( \frac{5 \pi}{6} \right) \), we refer to the unit circle and see that \( \cos \left( \frac{5 \pi}{6} \right) = -\frac{\sqrt{3}}{2} \).
- Similarly, for \( \sin \left( \frac{5 \pi}{6} \right) \), we find \( \sin \left( \frac{5 \pi}{6} \right) = \frac{1}{2} \).
Using these values, we substitute back into Euler's formula and then multiply by the magnitude 2 to find our rectangular components. This crucial step simplifies the process of converting from polar to rectangular form.