Problem 38 - Advanced Engineering Mathematics

Question

We have $$(\cos \theta+i \sin \theta)^{3}=\cos 3 \theta+i \sin 3 \theta$$ Also $$\begin{aligned} (\cos \theta+i \sin \theta)^{3} &=\cos ^{3} \theta+3 \cos ^{2} \theta(i \sin \theta)+3 \cos \theta(i \sin \theta)^{2}+(i \sin \theta)^{3} \\ &=\cos ^{3} \theta-3 \cos \theta \sin ^{2} \theta+\left(3 \cos ^{2} \theta \sin \theta-\sin ^{3} \theta\right) i \end{aligned}$$ Equating real and imaginary parts gives $$\cos 3 \theta=\cos ^{3} \theta-3 \cos \theta \sin ^{2} \theta, \quad \sin 3 \theta=3 \cos ^{2} \theta \sin \theta-\sin ^{3} \theta$$

Step-by-Step Solution

Answer

Identity confirmed: real and imaginary parts equate.

Step 1: Understand the Expression

The given equation is \((\cos \theta + i \sin \theta)^{3} = \cos 3\theta + i\sin 3\theta\). This expression uses complex numbers formatted as Euler's formula, combining trigonometric functions with imaginary numbers.

Step 2: Expand the Cubic Expression

Perform the expansion \((\cos \theta + i \sin \theta)^3\). This follows the binomial theorem: \[(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\] where \(a = \cos \theta\) and \(b = i \sin \theta\).

Step 3: Calculate Each Term

Calculate each term separately: - \(a^3 = \cos^3 \theta\)- \(3a^2b = 3 \cos^2 \theta (i \sin \theta) = 3i \cos^2 \theta \sin \theta\)- \(3ab^2 = 3 \cos \theta (i \sin \theta)^2 = -3 \cos \theta \sin^2 \theta\)- \(b^3 = (i \sin \theta)^3 = -i \sin^3 \theta\)

Step 4: Simplify the Expression

Combine the real and imaginary terms:- The real part: \(\cos^3 \theta - 3 \cos \theta \sin^2 \theta\)- The imaginary part: \(3 \cos^2 \theta \sin \theta - \sin^3 \theta\)\(i\)

Step 5: Compare with Euler's Formula

Align with Euler's formula: \(\cos 3\theta + i \sin 3\theta\), which gives:- Real part: \(\cos 3\theta = \cos^3 \theta - 3 \cos \theta \sin^2 \theta\)- Imaginary part: \(\sin 3\theta = 3 \cos^2 \theta \sin \theta - \sin^3 \theta\)

Key Concepts

Complex Numbers

Complex numbers are fascinating entities that extend our traditional number system to encompass numbers written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. The imaginary unit \( i \) is defined as the square root of -1, providing a foundation for working with negative square roots.

One significant aspect of complex numbers is their ability to express trigonometric values in exponential form, especially useful in calculations involving oscillatory functions. This connection was popularized by Euler's formula, which states:

• \( e^{i\theta} = \cos \theta + i \sin \theta \)

This formula seamlessly connects trigonometric functions with exponential growth and decay, providing a powerful mathematical bridge.

In the problem we reviewed, the equation \((\cos \theta + i \sin \theta)^3 = \cos 3\theta + i \sin 3\theta\) exemplifies the interaction of complex numbers represented in trigonometric form. Such representation makes calculations straightforward, particularly for powers and roots of complex numbers.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables, featuring the relationships among trigonometric functions. These identities are essential tools in simplifying expressions and solving trigonometric equations.

A crucial use of trigonometric identities in the given exercise involves the derivation of the triple angle identities for sine and cosine. After expanding \((\cos \theta + i \sin \theta)^3\) using the binomial theorem, the following identities are reached by equating the real and imaginary parts:

• Cosine Triple Angle: \( \cos 3\theta = \cos^3 \theta - 3\cos \theta \sin^2 \theta \)
• Sine Triple Angle: \( \sin 3\theta = 3\cos^2 \theta \sin \theta - \sin^3 \theta \)

These identities are essential in advancing your understanding of how trigonometric functions can be transformed and represented. They allow you to express functions of larger angles in terms of functions of smaller angles, easing calculations especially in calculus and higher mathematics.

Understanding these identities fosters a deeper grasp of how trigonometric functions interrelate, aiding in the solution of more complex problems.

Binomial Theorem

The binomial theorem is a powerful algebraic method to expand expressions raised to any positive integer power. This theorem is articulated as follows:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)

In simple terms, it provides a formula to expand expressions like \( (a + b)^3 \) systematically, by adding terms obtained through multiplication and taking powers.

Applying the binomial theorem to the expansion \((\cos \theta + i \sin \theta)^3\):

• First term: \( \cos^3 \theta \)
• Second term: \( 3 \cos^2 \theta (i \sin \theta) = 3i \cos^2 \theta \sin \theta \)
• Third term: \( 3 \cos \theta (i \sin \theta)^2 = -3 \cos \theta \sin^2 \theta \)
• Fourth term: \((i \sin \theta)^3 = -i \sin^3 \theta \)

This generates both the real and imaginary components necessary to compare the expanded expression to De Moivre's form \(\cos 3\theta + i\sin 3\theta\).

Mastering the binomial theorem not only aids in handling algebraic manipulations but also enhances problem-solving strategies in various branches of mathematics, including probability and calculus.