Problem 36 - Algebra and Trigonometry

Question

Write each complex number in rectangular form. $$ 3 e^{i \frac{\pi}{10}} $$

Step-by-Step Solution

Answer

2.85318 + 0.92706 i

Step 1: Recall Euler's Formula

Euler's Formula states that \footnote{\text{Euler's formula:}} \(e^{i \theta} = \text{cos}(\theta) + i \text{sin}(\theta)\). This formula will be used to convert the complex number to its rectangular form.

Step 2: Identify \theta and r

For the given complex number \text{(i.e.,} 3 e^{i \frac{\pi}{10}}\text{)}, identify the magnitude \( r = 3 \) and the angle \( \theta = \frac{\pi}{10} \).

Step 3: Apply Euler's Formula

Use Euler's formula: \(e^{i \frac{\pi}{10}} = \text{cos} \left( \frac {\pi}{10} \right) + i \text{sin} \left( \frac {\pi}{10} \right)\).

Step 4: Multiply by the magnitude

Multiply both parts of the complex number by the magnitude \( r = 3 \), \(3 e^{i \frac{\pi}{10}} = 3 \left[ \text{cos} \left( \frac {\pi}{10} \right) + i \text{sin} \left( \frac {\pi}{10} \right) \right]\).

Step 5: Calculate the values

Compute \( \text{cos} \left( \frac{\pi}{10} \right)\approx 0.95106 \) and \(\text{sin} \left( \frac{\pi}{10} \right)\approx 0.30902\) . Therefore: \( 3 \left[ 0.95106 + i 0.30902 \right]\).

Step 6: Distribute the magnitude

Distribute \( r = 3\) throughout the sum: \( 3 \times 0.95106 + 3 i \times 0.30902 = 2.85318 + 0.92706 i\).

Key Concepts

Euler's formula

One of the key concepts in understanding complex numbers is Euler's Formula. Euler's Formula provides a bridge between the exponential function and trigonometric functions. It states that for any real number \( \theta \):

\( e^{i \theta} = \text{cos}(\theta) + i \text{sin}(\theta) \).

This formula shows how complex exponentiation relates to circular (trigonometric) functions. By using it, we can convert from polar to rectangular (Cartesian) form of a complex number. Let's see how this works.

In polar form, a complex number is represented as \( r e^{i \theta} \), where \( r \) is the magnitude and \( \theta \) is the angle. Using Euler's formula, \( e^{i \theta} \) can be rewritten as \( \text{cos}(\theta) + i \text{sin}(\theta) \). So, \( r e^{i \theta} \) becomes \( r (\text{cos}(\theta) + i \text{sin}(\theta)) \). This captures both the magnitude and direction of the complex number.

Rectangular form

The rectangular (or Cartesian) form of a complex number is one of the two standard ways to represent complex numbers. It is written as \( a + bi \), where:

• \( a \) is the real part
• \( bi \) is the imaginary part

Each complex number can be viewed as a point or vector in a two-dimensional plane called the complex plane. In this form, we can simply plot the real part \( a \) on the x-axis and the imaginary part \( b \) on the y-axis.

For example, let's convert the complex number \(3 e^{i \frac{\theta}{10}}\) into rectangular form. Using Euler's formula, we get:
\( e^{i \frac{\theta}{10}} = \text{cos}(\theta/10) + i \text{sin}(\theta/10) \).
We then multiply both parts by the magnitude \( r = 3 \):
\( 3 \times (\text{cos}(\theta/10) + i \text{sin}(\theta/10)) \)=\( 3 \text{cos}(\theta/10) + 3 i \text{sin}(\theta/10) \).
This process yields the complex number in rectangular form: \( 2.85318 + 0.92706i \) as calculated in the step-by-step solution.

Magnitude and angle in polar form

When working with complex numbers, the polar form is another common representation. The polar form expresses a complex number in terms of its magnitude and angle from the positive real axis.

A complex number \( z = r e^{i \theta} \) is represented by:

• \( r \): The magnitude (or modulus) of the complex number, which is the distance from the origin to the point in the complex plane. It is given by \( r = |z| = \text{sqrt}(a^2 + b^2) \).
• \( \theta \): The angle (or argument) measured counterclockwise from the positive real axis to the line representing the complex number. It is typically expressed in radians, and calculations often use \( \theta = \text{atan2}(b, a) \).

To convert polar form to rectangular form, use Euler's formula: \( e^{i \theta} = \text{cos}(\theta) + i \text{sin}(\theta) \). So, \( z = r e^{i \theta} \) becomes \( r (\text{cos}(\theta) + i \text{sin}(\theta)) \).

For the conversion process: first, identify the magnitude \( r \) and the angle \( \theta \). Then apply Euler's formula by substituting \( \text{cos}(\theta) \) and \( i \text{sin}(\theta) \) with their respective values. Finally, multiply through by the magnitude \( r \) to get the rectangular form.