Problem 117 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[2(\cos \pi+i \sin \pi)]^{14}$$

Step-by-Step Solution

Answer

Therefore, the complex number \([2(\cos \pi+i \sin \pi)]^{14}\) in standard form is \(16384+0i\).

Step 1: Applying DeMoivre's Theorem

DeMoivre's Theorem states that \((r(\cos \Theta+i \sin \Theta))^n = r^n(\cos n\Theta +i \sin n\Theta)\). Applying this to our complex number, where \(r=2\), \(\Theta=\pi\), and \(n=14\), we get \((2(\cos \pi+i \sin \pi))^{14} = 2^{14}(\cos 14\pi +i \sin 14\pi)\).

Step 2: Simplification

We know that \(\cos x\) and \(\sin x\) are periodic functions with period \(2\pi\). They repeat their values every \(2\pi\). Thus \(\cos 14\pi = \cos 2\pi\), and \(\sin 14\pi = \sin 2\pi\), which both simplify to 1 and 0 respectively. Therefore, we have \((2(\cos \pi+i \sin \pi))^{14} = 2^{14}(1 +i*0)\).

Step 3: Converting to standard form (a+bi)

The standard form of a complex number is \(a+bi\). In our case, \(2^{14}\) simplifies to 16384. Therefore, \((2(\cos \pi+i \sin \pi))^{14} = 16384*(1+i*0) = 16384+0i.\)

Key Concepts

Complex Numbers

Complex numbers are mathematical constructs that combine real numbers with imaginary numbers. You might often see them written in the form \(a + bi\), where \a\ is the real part and \b\ is the imaginary part. The imaginary unit, \i\, satisfies the equation \(i^2 = -1\). In reality, complex numbers exist in a two-dimensional plane rather than the one-dimensional line that you're used to with real numbers.

• The real part of a complex number is similar to any regular number, like 3 or -4.5.
• The imaginary part involves the imaginary unit \i\, which is a special number that helps with calculating roots of negative numbers.

This kind of number is used in various fields such as engineering, physics, and applied mathematics to describe phenomena that cannot be explained with just real numbers. Complex numbers are fundamental when dealing with waves and periodic phenomena, like alternating current in electrical engineering.

Polar Form

Polar form is a way to express complex numbers using magnitudes and angles, rather than the standard rectangular form. You'll often see the polar form written as \(r(\cos \Theta + i\sin \Theta)\), where \r\ is the modulus and \Theta\ is the argument or angle. This is particularly handy when you're working with properties like angles and rotations.

• The modulus \r\ represents the distance from the origin to the point in the complex plane.
• The argument \Theta\ is the angle formed with the positive real axis.

Using polar form makes it easier to perform certain operations like multiplication and division, and it's especially useful for calculating powers and roots through DeMoivre’s Theorem. This is evident when you raise a complex number to a power or root, as multiplying or dividing moduli and adding or subtracting angles is far simpler in polar coordinates.

Standard Form

The standard form of a complex number is its most straightforward representation, usually written as \(a + bi\). It breaks down the complex number into its real and imaginary components. In this form, there is no need for angles or modulus; it is purely about the cartesian coordinates or points.

• Standard form provides a simple, direct way to understand the composition of a complex number.
• It makes addition and subtraction much simpler because you can just add or subtract real terms and imaginary terms separately.

When problems require you to find answers in standard form, they are looking for the result in everyday cartesian coordinates. For example, if you raise a complex number to a power using DeMoivre's Theorem, you might be asked to convert it back into standard form to easily communicate the result.

Exponential Form

Exponential form is a compact way to express complex numbers, relying heavily on Euler's formula. Euler's relation states \(e^{i\Theta} = \cos \Theta + i\sin \Theta\). This makes the exponential form \(re^{i\Theta}\), combining the modulus \r\ with the exponential term representing the angle.

• This form leverages the power of exponentials to simplify complex number multiplication and division.
• Euler's formula bridges trigonometric functions with the imaginary unit \i\, offering a seamless way to handle calculations involving rotations and periodic functions.

The exponential form is especially convenient for mathematicians and engineers working with oscillations and waves, as it simplifies differentiating and integrating periodic functions. It's a powerful tool that, when combined with DeMoivre's Theorem, makes handling powers and roots of complex numbers far more manageable.