Problem 59 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$3\left[\cos \left(\frac{11 \pi}{12}\right)+i \sin \left(\frac{11 \pi}{12}\right)\right]$$

Step-by-Step Solution

Answer

The rectangular form is approximately \(-0.7764 + 2.8977i\).

Step 1: Understanding the Problem

The complex number is given in polar form: \( 3[\cos(\frac{11\pi}{12}) + i\sin(\frac{11\pi}{12})] \). Our goal is to convert this into rectangular form \( a + bi \), where \( a \) and \( b \) are real numbers.

Step 2: Identify Components

Identify the radius \( r = 3 \) and the angle \( \theta = \frac{11\pi}{12} \). These correspond to the modulus and argument of the complex number in polar form.

Step 3: Calculate the Cosine Component

To find the real part of the rectangular form, calculate \( a = r \cos \theta = 3 \cos(\frac{11\pi}{12}) \). Use a calculator to find \( \cos(\frac{11\pi}{12}) \).

Step 4: Calculate the Sine Component

To find the imaginary part, calculate \( b = r \sin \theta = 3 \sin(\frac{11\pi}{12}) \). Use the calculator to find \( \sin(\frac{11\pi}{12}) \).

Step 5: Compute the Values

Using a calculator: 1. \( \cos(\frac{11\pi}{12}) \approx -0.2588 \), so \( a = 3 \times -0.2588 \approx -0.7764 \).2. \( \sin(\frac{11\pi}{12}) \approx 0.9659 \), so \( b = 3 \times 0.9659 \approx 2.8977 \).

Step 6: Write the Solution in Rectangular Form

Thus, the rectangular form of the complex number is \( a + bi \approx -0.7764 + 2.8977i \).

Key Concepts

Polar Form

In mathematics, the polar form represents a complex number in a visually intuitive way. Instead of using the standard rectangular coordinates, a complex number in polar form is expressed using a

• modulus, \(|z|\), which is the distance from the origin to the point;
• argument, \(\theta\), which is the angle formed with the positive real axis.

Polar form is like saying "how far" and "in which direction" the number is from the origin in the complex plane. For example, our given problem expresses the complex number as \(3[\cos(\frac{11\pi}{12}) + i\sin(\frac{11\pi}{12})]\). The number 3 is the modulus, and \(\frac{11\pi}{12}\) is the argument.The advantage of polar form arises in multiplying complex numbers. Multiplication is performed by simply multiplying the moduli and adding the angles. This makes handling complex arithmetic more manageable.

Rectangular Form

Rectangular form, also known as Cartesian form, is the standard format for writing complex numbers. While polar form uses angles and radii, rectangular form focuses only on horizontal and vertical components. This form follows the structure \(a + bi\), where:

• \(a\) is the real part;
• \(b\) is the imaginary part.

Think of rectangular form like ordinary coordinates on a graph, where each complex number has a location. For our exercise, converting from polar to rectangular form results in \(-0.7764 + 2.8977i\), which places our number in the complex plane using real and imaginary components.This form is beneficial for adding and subtracting complex numbers, as these operations align each component independently, much like dealing with ordinary vectors.

Conversion Between Forms

Understanding how to switch between polar and rectangular forms is essential in handling complex numbers effectively. The key is to know that:

• Polar form can be converted to rectangular form using the formulas: \(a = r\cos(\theta)\), \(b = r\sin(\theta)\).
• Rectangular form can be converted to polar form using: \(|z| = \sqrt{a^2 + b^2}\), \(\theta = \tan^{-1}(\frac{b}{a})\).

For instance, in our exercise, we started with polar form and used trigonometric functions to obtain \(a\) and \(b\). This conversion unveils the real and imaginary parts required for rectangular form and helps visualize how the complex number sits on a plane.

Trigonometric Functions

Trigonometric functions are vital in the conversion process between polar and rectangular forms. These functions allow us to translate angle information into useful segment lengths:

• \(\cos(\theta)\) translates the angle into the horizontal distance from the origin.
• \(\sin(\theta)\) translates the angle into the vertical distance from the origin.

For our conversion, we employ:\[ 3 \cos\left(\frac{11\pi}{12}\right) \] and \[ 3 \sin\left(\frac{11\pi}{12}\right) \] to determine the \(a\) and \(b\) components of the complex number. Using a calculator to find these values allows us to switch seamlessly between polar and rectangular perspectives, capturing both magnitude and directional properties.