Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 39-42: Prove that the statement is true for every positive integer \(n\). Prove De Moivre's theorem: $$ [r(\cos \theta+i \sin \theta)]^{n}=r^{n}(\cos n \theta+i \sin n \theta) $$ for every positive integer \(n\).

Step-by-Step Solution

Answer

De Moivre's theorem is true for every positive integer:\( [r(\cos \theta+i \sin \theta)]^{n}=r^{n}(\cos n \theta+i \sin n \theta) \).

Step 1: Understanding the Statement

De Moivre's theorem states that for a complex number expressed in polar form \( r(\cos \theta + i \sin \theta) \), raising it to the power of \( n \) yields \( r^{n}(\cos n\theta + i \sin n\theta) \). Our goal is to prove this for any positive integer \( n \).

Step 2: Base Case Verification

To begin the mathematical induction, verify the base case \( n = 1 \). Substitute \( n = 1 \) into the formula: \([r(\cos \theta+i \sin \theta)]^{1}=r^{1}\cos 1\theta+i \sin 1\theta\).Simplifying this, we get \(r(\cos \theta+i \sin \theta) = r(\cos \theta + i \sin \theta)\), which holds true, hence the base case is verified.

Step 3: Inductive Hypothesis

Assume that De Moivre's theorem is true for an arbitrary positive integer \( k \), i.e.,\([r(\cos \theta+i \sin \theta)]^{k}=r^{k}(\cos k \theta+i \sin k \theta)\).This assumption is what we call the inductive hypothesis.

Step 4: Inductive Step Proof

We need to show that the theorem holds true for \( k + 1 \). Using the inductive hypothesis:\([r(\cos \theta + i \sin \theta)]^{k + 1} = [r(\cos \theta + i \sin \theta)]^k \cdot [r(\cos \theta + i \sin \theta)]\).Substitute the inductive hypothesis:\(= r^k (\cos k\theta + i \sin k\theta) \cdot r(\cos \theta + i \sin \theta)\).

Step 5: Expanding the Expression

Expand the multiplication using distributive property:\[= r^{k+1} [(\cos k\theta \cos \theta - \sin k\theta \sin \theta) + i (\cos k\theta \sin \theta + \sin k\theta \cos \theta)]\]. Recall that:\(\cos(k\theta + \theta) = \cos k\theta \cos \theta - \sin k\theta \sin \theta\), \(i\sin(k\theta + \theta) = \cos k\theta \sin \theta + \sin k\theta \cos \theta\).

Step 6: Conclusion

Therefore,\[= r^{k+1} (\cos((k+1)\theta) + i \sin((k+1)\theta))\],which proves the statement for \( n = k+1 \). This completes the induction, and thus, De Moivre's theorem holds true for all positive integers \( n \).

Step 7: Final Verification

Having verified the base case and performed the inductive step, we conclude by the principle of mathematical induction that De Moivre's theorem is true for every positive integer \( n \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part. They can be written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part \(i\). The imaginary unit \(i\) is defined as \(\sqrt{-1}\). Complex numbers extend the concept of one-dimensional number lines to a two-dimensional complex plane.

• Real numbers are on the x-axis.
• Imaginary numbers are on the y-axis.

The combination provides a comprehensive way of representing numbers that can't be expressed on the real line alone.
They play a vital role in various fields of mathematics and engineering, particularly when grappling with solutions to equations that do not have real solutions.

Polar Form

The polar form is another way to express complex numbers, using the magnitude \(r\) and an angle \(\theta\). It provides a more intuitive understanding of a complex number's position and behavior, especially under operations like multiplication and exponentiation.

A complex number \( z = a + bi \) can be represented in polar form as \( r(\cos \theta + i\sin \theta) \), where:

• \(r\) is the modulus, calculated as \(\sqrt{a^2 + b^2}\).
• \(\theta\) is the argument, which can be found using \(\tan^{-1}(b/a)\).

Converting complex numbers to polar form makes it simpler to perform certain mathematical operations, such as finding powers and roots, as seen in De Moivre's theorem.
It emphasizes the importance of understanding angles and distances in the complex plane, rather than just horizontal (real) and vertical (imaginary) components.

Mathematical Induction

Mathematical induction is a proof technique commonly used to establish that a statement is true for all natural numbers. The process consists of two main steps:

• Base Case: Verify the statement is true for the initial value (typically \(n = 1\)). This serves as the foundation.
• Inductive Step: Assume the statement is true for \(n = k\) (inductive hypothesis), and prove that truth for \(n = k + 1\) logically follows. This makes the step from one case to the next.

This method forms a domino effect; if true for the initial case, and the truth of one case implies the truth of the next, it must be true for all cases. In proving De Moivre's theorem, induction shows that if the theorem holds for one power, it holds for the next, thereby confirming it for all positive integers.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities are instrumental in simplifying expressions and solving equations. Some commonly used identities include:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(\cos(a + b) = \cos a \cos b - \sin a \sin b\)
• \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)

In De Moivre's theorem, trigonometric identities help in expanding and simplifying the products of complex numbers in their polar form. They allow for the transition from the multiplication of trigonometric expressions to the sum of trigonometric angles, as seen in the inductive step of the proof.

Understanding and utilizing these identities are crucial in working with and solving problems involving complex numbers and their powers.