Question

\(5\left(\cos \frac{2 \pi}{5}+i \sin \frac{2 \pi}{5}\right)\)

Short answer

The expression can be rewritten as \(5 e^{i \frac{2 \pi}{5}}\). The approximate values are 1.545 (real part) and 4.755 (imaginary part).

Step-by-step solution

- Understand the Given Expression

The given expression is in the form of a complex number, represented in polar form: \[5\left(\cos \frac{2 \pi}{5} + i \sin \frac{2 \pi}{5}\right)\]

- Recall Euler's Formula

Euler's formula states that \( e^{i \theta} = \cos \theta + i \sin \theta \). Compare this with the given expression.

- Rewrite Using Euler's Formula

Rewrite the expression using Euler's formula: \[5\left(\cos \frac{2 \pi}{5} + i \sin \frac{2 \pi}{5}\right) = 5 e^{i \frac{2 \pi}{5}}\]

- Identify the Angle

The angle is \( \frac{2 \pi}{5} \) which falls within the principal argument range (-π, π].

- Calculate the Real and Imaginary Parts

Express the real and imaginary parts: The real part is: \[5 \cos \frac{2 \pi}{5}\] The imaginary part is: \[5 \sin \frac{2 \pi}{5}\] Use the unit circle to find the values.

- Numerical Approximation (Optional)

Approximate the constants \(\cos \frac{2 \pi}{5}\) and \(\sin \frac{2 \pi}{5}\): \[\cos \frac{2 \pi}{5} \approx 0.309 \] \[\sin \frac{2 \pi}{5} \approx 0.951\] Hence, \[5 \cos \frac{2 \pi}{5} \approx 5 \times 0.309 = 1.545\] \[5 \sin \frac{2 \pi}{5} \approx 5 \times 0.951 = 4.755\]

Key concepts

Euler's formula

Euler's formula is a fundamental bridge in mathematics connecting complex numbers and trigonometry. It states that for any real number \theta, we have: \( e^{i \theta} = \text{cos} \, \theta + i \, \text{sin} \, \theta \).
In simpler terms, Euler's formula tells us how to represent complex numbers in exponential form. This can make many calculations easier.
Understanding this formula is key to converting between polar and rectangular forms of complex numbers. In our exercise, we rewrite the polar form complex number using Euler's formula as \( 5 e^{i \frac{2 \pi}{5}} \). This highlights the power of Euler's formula in simplifying complex expressions.

Polar coordinates

Polar coordinates are a way of representing points in terms of their distance from a reference point and the angle from a reference direction. Instead of using \( (x, y) \) like in Cartesian coordinates, we use \( (r, \theta) \), where:

• \(r\) is the distance from the origin.
• \(\theta\) is the angle measured from the positive x-axis.

In our exercise, the complex number is given in polar form as \(5\bigg(\text{cos} \, \frac{2 \pi}{5} + i \, \text{sin} \, \frac{2 \pi}{5}\bigg)\). Here, 5 is the magnitude (or radius), and \(\frac{2\pi}{5}\) is the angle.

Real and imaginary parts computation

To compute the real and imaginary parts of a complex number from its polar form, we use the trigonometric functions:

• The real part is given by \(r \, \text{cos} \, \theta\).
• The imaginary part is given by \(r \, \text{sin} \, \theta\).

In our example, the number \(5 \bigg(\text{cos} \, \frac{2 \pi}{5} + i \, \text{sin} \, \frac{2 \pi}{5}\bigg)\) translates to:

• Real part: \(5 \, \text{cos} \, \frac{2 \pi}{5}\)
• Imaginary part: \(5 \, \text{sin} \, \frac{2 \pi}{5}\)

By substituting the angle into the trigonometric functions, we can find these parts precisely.

Unit circle

The unit circle is a circle of radius 1 centered at the origin of the coordinate plane. It's a powerful tool to visualize and understand trigonometric functions and angles. Each point on the unit circle corresponds to a complex number of the form \( \text{cos} \, \theta + i \, \text{sin} \, \theta \).
In our exercise, we can visualize the angle \( \frac{2 \pi}{5} \) on the unit circle, making it easier to appreciate why \( \text{cos} \, \frac{2 \pi}{5} \) and \( \text{sin} \,\frac{2 \pi}{5} \) are specific fixed values. The real part is the x-coordinate, and the imaginary part is the y-coordinate on this circle.

Numerical approximation

Sometimes, exact values of trigonometric functions result in cumbersome numbers. Hence, we often use numerical approximations for simplicity.
For this example, we approximate:

• \( \text{cos} \, \frac{2 \pi}{5} \approx 0.309 \)
• \( \text{sin} \, \frac{2 \pi}{5} \approx 0.951 \)

Using these approximations, we get:

• Real part: \( 5 \, \text{cos} \, \frac{2 \pi}{5} \approx 5 \times 0.309 = 1.545 \)
• Imaginary part: \( 5 \, \text{sin} \, \frac{2 \pi}{5} \approx 5 \times 0.951 = 4.755 \)

These approximations make it easier to handle and interpret complex numbers in practical scenarios.