Problem 64 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$6\left[\cos \left(\frac{13 \pi}{8}\right)+i \sin \left(\frac{13 \pi}{8}\right)\right]$$

Step-by-Step Solution

Answer

The rectangular form is approximately \(-2.298 + 5.544i\).

Step 1: Understand the Problem

We need to convert the given complex number in trigonometric form to rectangular form. The trigonometric form is given as \( r(\cos(\theta) + i \sin(\theta)) \), and we need to find the rectangular form \( a + bi \). Here, \( r = 6 \) and \( \theta = \frac{13\pi}{8} \).

Step 2: Calculate \( \cos\left(\frac{13\pi}{8}\right) \)

Use a calculator to find \( \cos\left(\frac{13\pi}{8}\right) \). Inputting \( \cos(2.042) \) (approximately), we find that \( \cos\left(\frac{13\pi}{8}\right) \approx -0.383 \).

Step 3: Calculate \( \sin\left(\frac{13\pi}{8}\right) \)

Similarly, use a calculator to find \( \sin\left(\frac{13\pi}{8}\right) \). Inputting \( \sin(2.042) \) (approximately), we find that \( \sin\left(\frac{13\pi}{8}\right) \approx 0.924 \).

Step 4: Multiply by the Magnitude

Multiply both the cosine and sine values by the magnitude \( r = 6 \). So, calculate: \[-0.383 \times 6 = -2.298\]and \\[0.924 \times 6 = 5.544\].

Step 5: Write in Rectangular Form

Combine the results to express the complex number in rectangular form:\(-2.298 + 5.544i\).

Key Concepts

Rectangular Form

Rectangular form is a way of expressing complex numbers that is easy to visualize, especially on a graph. In rectangular form, complex numbers are written as \( a + bi \), where \( a \) represents the real part, and \( bi \) represents the imaginary part. The components \( a \) and \( b \) are real numbers, while \( i \) is the imaginary unit, having the property \( i^2 = -1 \).

• The real part \( a \) gives the number's position on the horizontal axis.
• The imaginary part \( b \) determines the position on the vertical axis.

Imagine plotting this on a Cartesian plane, where the horizontal line (x-axis) deals with real numbers, and the vertical line (y-axis) with imaginaries. The point \( (a, b) \) will help you visually assess the complex number's place.
The beauty of rectangular form lies in its straightforward application in addition, subtraction, and providing an easy way to calculate magnitude and conjugates of complex numbers.

Trigonometric Form

Trigonometric form (also referred to as polar form) is another method for expressing complex numbers. In this form, a complex number is represented as \( r(\cos(\theta) + i \sin(\theta)) \), where:

• \( r \) is the modulus or magnitude of the complex number.
• \( \theta \) is the argument or angle the line makes with the positive real axis.

This form is particularly beneficial in complex calculations involving multiplication and division, as it utilizes the properties of trigonometric functions. Here:

• The modulus \( r \) equals the distance from the origin to the point \( (a, b) \) and is calculated as \( \sqrt{a^2 + b^2} \).
• The argument \( \theta \) is the angle measured from the positive x-axis to the line connecting the origin and the point \( (a, b) \), typically evaluated using \( \theta = \tan^{-1}(\frac{b}{a}) \).

Using trigonometric form simplifies the process when dealing with powers of complex numbers, working with roots, or even converting back to rectangular form.

Conversion

Conversion between rectangular form and trigonometric form is a crucial skill in working with complex numbers. To move from trigonometric to rectangular form, as you do with the exercise, involves these steps:
1. **Identify the modulus** \( r \) and argument \( \theta \) from the trigonometric form, \( r(\cos(\theta) + i \sin(\theta)) \).
2. **Calculate real part**: Multiply \( \cos(\theta) \) by \( r \) to find \( a \).
3. **Calculate imaginary part**: Multiply \( \sin(\theta) \) by \( r \) for \( b \).
4. **Combine**: Express the result in \( a + bi \) form.

Conversely, to convert from rectangular to trigonometric form:

• **Find the magnitude**: Use \( r = \sqrt{a^2 + b^2} \).
• **Determine the angle**: Use \( \theta = \tan^{-1}(\frac{b}{a}) \) to find the argument.
• Express the number in \( r(\cos(\theta) + i \sin(\theta)) \).

Having these conversion skills at your disposal will unlock a deeper understanding of how complex numbers function in different mathematical contexts.