Problem 113 - 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. $$\left[5\left(\cos 140^{\circ}+i \sin 140^{\circ}\right)\right]^{3}$$

Step-by-Step Solution

Answer

The power of the complex number is \(62.5 + 108.253175i\).

Step 1: Apply DeMoivre's Theorem

DeMoivre's theorem dictates that when a complex number in trigonometric form, such as \(r(\cos \theta + i \sin \theta)\), is raised to a power \(n\), it can be simplified to: \(r^n(\cos n\theta + i \sin n\theta)\). So, here the complex number is \(5(\cos 140^{\circ}+i \sin 140^{\circ})\) and \(n=3\). Therefore, this simplifies to: \(5^3(\cos(3*140^{\circ}) + i \sin(3*140^{\circ}))\).

Step 2: Simplify the complex number

Simplifying gets \(125[\cos(420^{\circ}) + i \sin(420^{\circ})]\). This is because \(5^3=125\) and \(140^{\circ}*3 = 420^{\circ}\).

Step 3: Convert to standard form

A complex number in trigonometric form, \(r(\cos \theta + i \sin \theta)\), can be converted to standard form, \(a+bi\), with \(a = r\cos \theta\) and \(b = r\sin \theta\). Using our values, this results in \(125\cos(420^{\circ}) + 125i \sin(420^{\circ})\). However, because \(420^{\circ}\) is equivalent to \(60^{\circ}\), this simplifies to \(125\cos(60^{\circ}) + 125i \sin(60^{\circ})\), which finally results in \(62.5 + 108.253175i\).

Key Concepts

Complex Numbers

Complex numbers appear as expressions of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). Complex numbers are an extension of the real numbers and encompass a vast range of possible values.

They have applications across mathematics, engineering, physics, and other sciences for the description and analysis of waveforms, quantum mechanics, and much more.

• The real part \(a\) represents the horizontal location on the complex plane.
• The imaginary part \(bi\) represents the vertical location on the complex plane.
• Complex numbers can be added, subtracted, multiplied, and divided, following rules similar to real numbers, while incorporating the property of \(i\).

Understanding complex numbers is essential for advanced scientific and mathematical studies, as they provide a powerful tool for solving equations that have no real solutions.

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number is particularly useful for performing operations like multiplication and division and finding powers or roots. Any complex number \(a + bi\) can be expressed in this form as \(r(\cos \theta + i \sin \theta)\). This is also known as the polar form.

• The modulus \(r\) is calculated as \(r = \sqrt{a^2 + b^2}\) and represents the distance from the origin to the point \((a, b)\) on the complex plane.
• The angle \(\theta\), known as the argument, is given by \(\theta = \tan^{-1}(\frac{b}{a})\) and measures the angle formed with the positive real axis.

DeMoivre's Theorem is especially convenient when dealing with powers and roots in trigonometric form. It states that \([r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta)\), simplifying calculations significantly compared to using standard form.

Standard Form Conversion

To work effectively with functions or solve equations in the complex plane, it's often necessary to convert complex numbers from their trigonometric form back to the standard form \(a + bi\). This conversion requires calculating the real and imaginary components using trigonometric functions.

Given a complex number \(r(\cos \theta + i \sin \theta)\):

• The real part, \(a\), is found with \(a = r \cos \theta\).
• The imaginary part, \(b\), is determined by \(b = r \sin \theta\).

After utilizing DeMoivre’s theorem for powers and simplifying any multiple angles, one can then apply these formulas to achieve the standard form. In our exercise, this led from a trigonometric expression \(125(\cos 420^{\circ} + i \sin 420^{\circ})\) to its equivalent standard form \(62.5 + 108.253175i\) by recognizing that \(420^{\circ}\) reduces to \(60^{\circ}\), alongside using known cosine and sine values for \(60^{\circ}\).