Problem 71 - Precalculus with Limits

Question

Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[5\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)\right]^{3}$$

Step-by-Step Solution

Answer

The power of the given complex number is \(62.5 + 108.25i\).

Step 1: Identify the Complex Number and the Power

We have the complex number in the form \(r(cos(\theta) + isin(\theta))\), where \(r = 5\), \(\theta = 20^\circ\), and the power \(n = 3\).

Step 2: Apply DeMoivre's Theorem

According to DeMoivre's Theorem, \([r(cos(\theta) + isin(\theta))]^n = r^n(cos(n\theta) + isin(n\theta))\). Substituting our values, we get \(5^3 (cos(3*20^\circ) + isin(3*20^\circ))\).

Step 3: Simplify the Expression

By simplifying, we get \(125(cos(60^\circ) + isin(60^\circ))\).

Step 4: Convert Back to Standard Form

We can convert the result back to standard form using the identities \(cos(60^\circ)=1/2\) and \(sin(60^\circ)=\sqrt{3}/2\). So, we get \(125[(1/2) + i(\sqrt{3}/2)] = 62.5 + 108.25i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as the square root of \(-1\), meaning \(i^2 = -1\).
Complex numbers might seem abstract at first, but they're actually very useful in many areas of mathematics, especially in algebra and calculus.

• The real part \(a\) can be any real number.
• The imaginary part \(b\) involves the imaginary unit \(i\).
• Complex numbers can be added, subtracted, multiplied, and divided.

Understanding complex numbers gives you the tools to solve equations that don’t have real solutions, such as quadratic equations with negative discriminants.

Standard Form

Standard form of a complex number refers to its customary algebraic expression \(a + bi\). It's similar to writing numbers in decimal form for clarity and ease of understanding.
Writing complex numbers in standard form is useful when you need to perform basic arithmetic operations or when you want to present the numbers clearly.

• Helps in visualizing the numbers on a complex plane.
• Makes addition and subtraction straightforward: combine like terms (real with real and imaginary with imaginary).
• Important for conversion between different forms, such as polar coordinates and standard form.

By expressing complex numbers in this form, we can easily interpret their components in terms of real and imaginary parts.

Polar Coordinates

Polar coordinates offer a different way to represent complex numbers using a radius and angle, rather than real and imaginary components. A complex number in polar form looks like \(r(\cos(\theta) + i\sin(\theta))\), where \(r\) is the magnitude (or length) of the vector, and \(\theta\) is the angle (or direction) in degrees or radians.
Polar coordinates are significant because they simplify the multiplication and division of complex numbers. Instead of manipulating two pairs of numbers, you adjust the magnitude and add or subtract angles.

• Makes it easy to raise complex numbers to powers using DeMoivre's Theorem.
• Provides a clear geometric interpretation: \(r\) is the distance to the origin, and \(\theta\) is the angle from the positive real axis.
• Facilitates conversions between forms: by using trigonometric identities or Pythagoras' theorem, you can convert between polar and standard forms.

Understanding polar coordinates is essential for topics in trigonometry and complex analysis, where the behavior of functions with complex numbers is explored.