Question

First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. $$5\left(\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}\right)$$

Short answer

The complex number is given in polar forms: $$5(\text{cos} \frac{2 \theta}{5}) +i(\text{sin}\frac{2i \theta}{5})$$; /Graph Coordinates are: 5e^{i\frac{2}{5}\theta}

Step-by-step solution

Identify the Given Form

The problem gives the number in polar form: i.e., $$5\big(\text{cos} \frac{2\theta}{5} + \text{i sin} \frac{2\theta}{5}\big)$$

Convert to Rectangular Form

Using Euler's formula, $$re^{i \theta} = r(\text{cos}\theta + i \text{sin}\theta),$$ the given expression can be rewritten directly from its polar form: $$5\big(\text{cos} \frac{2 \theta}{5} + \text{i sin} \frac{2 \theta}{5}\big)$$ into $$5e^{i \frac{2}{5}\theta}.$$

Use Specific Values

Here, \(r = 5\) and $$\theta = \frac{2}{5}\theta.$$ In situations where $$ \theta = 2\theta,$$ use the given additional information above substituting in finding the as above directly in a simpler manner.

Plot in the Complex Plane

Graphically plot the following coordinates in the Complex plane on x (real) - y (imaginary) as such in 2-dimensional ways: Real part \(x = 5 \big(\text{cos} \frac{2 \theta}{5}(r)\big)\) Imaginary part \(y = 5 \big(\text{i sin} \frac{2i \theta}{5}(i)\big). \) Graph Coordinate: (Real, Imaginary)

Key concepts

polar form

Complex numbers can be represented in various ways, and one of the most elegant representations is the polar form. In polar form, a complex number is expressed as:
\(r(\text{cos}\theta + i \text{sin}\theta)\)
where \(r\) is the magnitude of the complex number, and \( \theta \) (theta) is the angle formed with the positive real axis—also known as the argument of the complex number. In the given problem, the complex number is provided as:
\(5 \big(\text{cos} \frac{2 \theta}{5} + i \text{sin} \frac{2 \theta}{5}\big)\)
Here, \(r = 5\) and \(\theta = \frac{2 \theta}{5}\). It's essential to know that this representation makes it simpler to perform multiplication and division of complex numbers.

rectangular form

The rectangular form of a complex number expresses it as \(x + iy\), where \(x\) is the real part and \(y\) is the imaginary part. To convert from polar form to rectangular form, we use the relations:
\( x = r \text{cos}(\theta) \)
\( y = r \text{sin}(\theta) \)
For example, in the given problem, where \(r = 5\) and \(\theta = \frac{2 \theta}{5}\), the real and imaginary parts of the rectangular form can be calculated as:

• Real part \(x = 5 \text{cos}\big(\frac{2 \theta}{5}\big)\)


• Imaginary part \(y = 5 \text{sin}\big(\frac{2 \theta}{5}\big)\)

Converting between forms helps visualize and perform operations with complex numbers efficiently.

Euler's formula

Euler's formula connects complex exponentials with trigonometric functions and is expressed as:
\( e^{i \theta} = \text{cos} \theta + i \text{sin} \theta \)
This powerful relationship allows us to write any complex number in polar form as:
\( re^{i \theta} = r(\text{cos} \theta + i \text{sin} \theta) \)
In the given exercise, the formula is applied by recognizing the expression:
\(5 \big(\text{cos} \frac{2 \theta}{5} + i \text{sin} \frac{2 \theta}{5}\big) = 5e^{i \frac{2 \theta}{5}}\)
By using Euler's formula, we transform the trigonometric representation into a succinct exponential form, which simplifies many calculations, especially in higher-level mathematics and engineering.

complex plane plotting

Plotting complex numbers on the complex plane provides a visual representation of these numbers and aids in understanding their properties. The complex plane consists of a horizontal axis (the real axis, representing real parts) and a vertical axis (the imaginary axis, representing imaginary parts). To plot the given number \(5 \big(\text{cos} \frac{2 \theta}{5} + i \text{sin} \frac{2 \theta}{5}\big)\):

• First determine the rectangular coordinates using \(x = 5 \text{cos} \big(\frac{2 \theta}{5}\big)\) and \(y = 5 \text{sin} \big(\frac{2 \theta}{5}\big)\)


• Next, mark the point corresponding to these coordinates on the complex plane as (\(x, y\))

This visualization assists in grasping the impact of the angle and magnitude on the position of the complex number and can be particularly useful for operations like addition, subtraction, and even more complex transformations.