Problem 2 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank. ______ Theorem states that if \(z=r(\cos \theta+i \sin \theta)\) is a complex number and \(n\) is a positive integer, then \(z^{n}=r^{n}(\cos n \theta+i \sin n \theta)\).

Step-by-Step Solution

Answer

De Moivre's Theorem

Step 1: Read the Statement

We need to fill in the blank in the given statement.

Step 2: Recall the Relevant Definition

Based on the context of the statement, we identify the correct mathematical term or concept that completes it.

Step 3: Complete the Statement

\(De Moivre's Theorem\)

Key Concepts

Complex Numbers

Complex numbers are a type of number that includes both a real part and an imaginary part. They are usually written in the form \( a + bi \), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \( \sqrt{-1} \).
Complex numbers extend the concept of one-dimensional numbers to two dimensions, allowing for richer algebraic and geometric representations.

• Real numbers lie along the horizontal axis.
• Imaginary numbers occupy the vertical axis.

In the complex plane, every complex number corresponds to a unique point with coordinates \((a, b)\). This representation makes complex numbers useful for solving algebraic equations that cannot be solved with real numbers alone.

Polar Form

In polar form, a complex number is expressed in terms of its magnitude and angle relative to the positive real axis. This can be particularly helpful for multiplying and dividing complex numbers.
To convert a complex number from its standard form \( a + bi \) to polar form \( r(\cos \theta + i \sin \theta) \):

• Calculate the magnitude \( r = \sqrt{a^2 + b^2} \).
• Find the angle \( \theta = \tan^{-1}(\frac{b}{a}) \), ensuring you place the angle in the correct quadrant based on the signs of \(a\) and \(b\).

Polar form simplifies many calculations, especially when applying De Moivre's Theorem for finding powers and roots of complex numbers.

Trigonometric Form

The trigonometric form of a complex number emphasizes its representation using trigonometric functions. It closely resembles the polar form, showing the complex number as:\[ z = r(\cos \theta + i \sin \theta) \]Here, \(r\) is the distance from the origin or the magnitude, and \(\theta\) is the angle, or argument, from the positive x-axis. Bringing trigonometric functions into the mix allows us to:

• Rotate the number around the origin by adjusting \(\theta\).
• Scale the number by multiplying by \(r\).

When looking at the trigonometric form, it becomes easy to use De Moivre's Theorem, which helps in raising the complex number to any integer power by simply multiplying the angle by the power and raising the magnitude to that power.

Powers of Complex Numbers

Raising complex numbers to a power can be cumbersome in the standard form but becomes much simpler using their trigonometric or polar form. De Moivre's Theorem provides a straightforward approach:Given a complex number \( z = r(\cos \theta + i \sin \theta) \), its power \(n\) is:\[ z^n = r^n (\cos(n\theta) + i\sin(n\theta)) \]This equation shows how the magnitude raises to the power \(n\), and the angle multiplies by \(n\). By converting complex numbers to polar or trigonometric form, computations of powers:

• Become algebraically easier.
• Enable you to visualize rotations and scaling in the complex plane.

Thus, De Moivre's Theorem is powerful when dealing with powers and roots of complex numbers.