Problem 31 - Algebra and Trigonometry

Question

Write each complex number in rectangular form. $$ 7 e^{i \pi} $$

Step-by-Step Solution

Answer

-7

Step 1: Understand the Given Form

The expression given, \( 7 e^{i \pi} \), is in polar form. The general polar form of a complex number is \( re^{i \theta} \), where \(r\) is the magnitude and \(\theta\) is the angle.

Step 2: Identify the Components

In \( 7 e^{i \pi} \), the magnitude \( r \) is 7, and the angle \( \theta \) is \( \pi \).

Step 3: Convert to Rectangular Form Using Euler's Formula

Euler's formula states that \( e^{i \theta} = \cos(\theta) + i\sin(\theta) \). Substitute \( \theta = \pi \) into the formula: \[ e^{i \pi} = \cos(\pi) + i\sin(\pi) \]Calculate the cosine and sine values: \[\cos(\pi) = -1 \quad \text{and} \quad \sin(\pi) = 0 \]So,\[ e^{i \pi} = -1 + 0i \]

Step 4: Multiply by the Magnitude

Multiply the result by the magnitude, 7: \[ 7 \cdot ( -1 + 0i ) = -7 + 0i \]Therefore, the rectangular form is \[ -7 \]

Key Concepts

rectangular form

When dealing with complex numbers, one common way to represent them is in rectangular form. A complex number in rectangular form appears as a + bi, where 'a' is the real part, and 'b' is the imaginary part. For example, if you have the complex number 3 + 4i, '3' is the real part, and '4i' is the imaginary part. In the realm of mathematics and engineering, this form helps in performing addition, subtraction, and multiplication of complex numbers. It is straightforward and visually intuitive as it matches the x-y plane layout. For our exercise, we started with a complex number in polar form, which needed to be converted to rectangular form using Euler's formula and basic trigonometric evaluations.

polar form

Complex numbers can also be represented in polar form. The general polar form of a complex number is written as re^(iθ), where 'r' is the magnitude (or modulus) of the complex number, and 'θ' is the angle (or argument) in radians. The polar form is particularly useful in scenarios such as signal processing and control systems because it elegantly combines magnitude and direction information. For the given problem, we started with 7e^(iπ), where 7 is the magnitude and π is the angle. The concept of polar coordinates aligns closely with how we visualize circular and rotational phenomena.

Euler's formula

Euler's formula is a fundamental equation in complex analysis, given by e^(iθ) = cos(θ) + i*sin(θ). This powerful formula bridges exponential functions and trigonometric functions with imaginary exponents. Named after the Swiss mathematician Leonhard Euler, the formula makes it possible to switch seamlessly between polar and rectangular forms of complex numbers. In our example, to convert 7e^(iπ) to rectangular form, we utilized Euler's formula by substituting θ = π. The equation simplifies as follows: e^(iπ) = cos(π) + i*sin(π) = -1 + 0i. We then multiplied this result by the magnitude 7, converting the polar form into the rectangular form -7.