Problem 60 - Precalculus

Question

Use a calculator to express each complex number in rectangular form. $$2\left[\cos \left(\frac{4 \pi}{7}\right)+i \sin \left(\frac{4 \pi}{7}\right)\right]$$

Step-by-Step Solution

Answer

The rectangular form is approximately \(-0.445 + 1.9498i\).

Step 1: Identify the given polar form

The complex number is given in polar form as \( 2\left[\cos \left(\frac{4 \pi}{7}\right) + i \sin \left(\frac{4 \pi}{7}\right)\right] \). In this format, \( 2 \) is the modulus and \( \frac{4\pi}{7} \) is the argument.

Step 2: Use the cosine formula

Calculate the real part by finding \( 2 \times \cos \left( \frac{4 \pi}{7} \right) \). Input \( \frac{4 \pi}{7} \) into your calculator to find the cosine value.

Step 3: Use the sine formula

Calculate the imaginary part by finding \( 2 \times \sin \left( \frac{4 \pi}{7} \right) \). Again, use your calculator to find the sine value for this argument.

Step 4: Substitute and compute using a calculator

Using a calculator, we find:- \( \cos \left(\frac{4 \pi}{7}\right) \approx -0.2225 \)- \( \sin \left(\frac{4 \pi}{7}\right) \approx 0.9749 \)Therefore, multiply:- Real part: \( 2 \times -0.2225 \approx -0.445 \)- Imaginary part: \( 2 \times 0.9749 \approx 1.9498 \)

Key Concepts

Polar Form of Complex Numbers

The polar form of complex numbers offers a unique way to represent complex numbers. In this format, a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the angle (or argument).
This form is extremely useful for understanding and performing operations like multiplication and division on complex numbers because it emphasizes their geometric interpretation.

• Magnitude (\( r \)): It's the distance from the origin to the point in the complex plane.
• Argument (\( \theta \)): It's the angle made with the positive real axis.

By converting a complex number from its rectangular form, \( a + bi \), to its polar form, computations can become more straightforward. This is partly because multiplying and dividing complex numbers in polar form involves only their magnitudes and the addition or subtraction of their arguments.

Complex Numbers

Complex numbers are an extension of the real numbers, integrating a real part and an imaginary part. A typical complex number is written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, while \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
They are incredibly useful in various fields like physics, engineering, and mathematics for analyzing problems involving two-dimensional movement.

• Real Part (\( a \)): The component of the complex number not associated with \( i \).
• Imaginary Part (\( bi \)): The component associated with \( i \).

This dual nature of complex numbers opens them up to calculations in both algebraic and geometric contexts, making them a vital component of advanced mathematical concepts. Their ability to represent oscillations, rotations, and other two-dimensional movements makes them indispensable in science and engineering.

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers is essentially the same as their polar form. When a complex number is written in this form, it beautifully combines magnitude and direction.
It employs trigonometric functions, cosine and sine, to establish its position in the complex plane.

• The cosine component, \( \cos \theta \), dictates the horizontal projection.
• The sine component, \( \sin \theta \), determines the vertical projection.

The trigonometric form is particularly advantageous when dealing with periodic functions and whenever transformations involving rotations and scaling are needed. It underscores the periodic nature of complex numbers and their ability to elegantly handle both amplitude and phase in calculations.