Problem 2 - Contemporary Precalculus

Question

Calculate the given product and express your answer in the form \(a+b i\). $$\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)^{20}$$

Step-by-Step Solution

Answer

Question: Calculate and express in the form \(a + bi\), where \(a\) and \(b\) are real numbers, the following complex number expression: \(\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)^{20}\). Answer: 1

Step 1: Apply De Moivre's theorem

We need to apply De Moivre's theorem to the given expression: $$\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)^{20} = \cos \left(20\cdot\frac{\pi}{5}\right) + i \sin \left(20\cdot\frac{\pi}{5}\right)$$

Step 2: Calculate \(20x\)

Let's find the product of \(20\) and \(x = \frac{\pi}{5}\): $$20 \cdot \frac{\pi}{5} = \frac{20\pi}{5} = 4\pi$$

Step 3: Evaluate cosine and sine functions

Now, evaluate the cosine and sine functions for the angle \(4\pi\): $$\cos 4\pi = 1$$ $$\sin 4\pi = 0$$

Step 4: Write the final answer

Putting our expressions together, the given expression becomes: $$\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)^{20} = 1 + 0i = 1$$ Hence, the final answer is \(1\).

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part with \(i\) representing the imaginary unit. The imaginary unit \(i\) is defined as \(i^2 = -1\).

This concept helps to extend the real number system to include solutions to problems that cannot be solved using only real numbers. In algebra, complex numbers allow us to find roots of equations and perform operations that are not possible with just real numbers.

When working with complex numbers, remember:

• Real numbers are a special case of complex numbers where the imaginary part is zero (\(b = 0\)).
• Complex conjugate pairs involve changing the sign of the imaginary part, which can be useful for simplification.

They play a crucial role in various fields, including engineering, physics, and mathematics, as they provide a more comprehensive understanding of solutions and behaviors in many contexts.

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers is a way to express complex numbers using trigonometric functions like sine and cosine. This form is particularly useful when dealing with powers and roots of complex numbers.

A complex number in trigonometric form is represented as \(r(\cos \theta + i\sin \theta)\), where \(r\) is the modulus (or magnitude) of the complex number, and \(\theta\) is the argument (or angle).

Here's how it works:

• \(r = \sqrt{a^2 + b^2}\): The modulus \(r\) is the distance from the origin to the point \((a, b)\) in the complex plane.
• \(\theta = \tan^{-1}(b/a)\): The argument \(\theta\) is the angle from the positive real axis to the line representing the complex number.

This form allows easy multiplication and division of complex numbers by addressing expressions like \((r_1r_2)[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]\) for products and uses the power of angles interpretation which aligns perfectly with De Moivre's Theorem.

Euler's Formula

Euler's formula is a key relationship in mathematics that connects complex numbers, trigonometry, and exponential functions. It is expressed as \(e^{i\theta} = \cos \theta + i\sin \theta\).

This formula beautifully links these three domains and is pivotal in converting complex numbers from exponential to trigonometric form. It simplifies the work with powers and roots, especially when paired with De Moivre's Theorem.

Key points to remember:

• When \(\theta = \pi\), Euler's formula results in the famous equation \(e^{i\pi} + 1 = 0\), known as Euler's identity.
• This formula demonstrates that exponential growths can be analyzed using periodic trigonometric functions, which is extraordinary in mathematics and physics.

Euler's formula broadens our ability to calculate complex expressions and provides a profound insight into how different mathematical concepts are intertwined. It enriches our understanding of complex systems and helps visualize complex numbers on a unit circle, making manipulation of these numbers more intuitive.