Problem 91 - Precalculus

Question

In advanced calculus, complex numbers in polar form are used extensively. Use De Moivre's formula to show that $$\cos (3 \theta)=4 \cos ^{3} \theta-3 \cos \theta$$

Step-by-Step Solution

Answer

Using De Moivre's Theorem, it is proven that \(\cos(3\theta) = 4\cos^3\theta - 3\cos\theta\).

Step 1: Understand De Moivre's Theorem

De Moivre's Theorem states that for any real number \(\theta\) and any integer \(n\), \((\cos\theta + i \sin\theta)^n = \cos(n\theta) + i \sin(n\theta)\). We'll use this theorem to expand \((\cos\theta + i \sin\theta)^3\).

Step 2: Expand \((\cos\theta + i \sin\theta)^3\) Using Binomial Theorem

Applying the Binomial Theorem to \((\cos\theta + i \sin\theta)^3\):\[(\cos\theta + i \sin\theta)^3 = \cos^3\theta + 3 i \cos^2\theta \sin\theta - 3 \cos\theta \sin^2\theta - i \sin^3\theta\]

Step 3: Simplify Using Trigonometric Identities

Rewrite \(\cos^2\theta\) and \(\sin^2\theta\) using \(\cos^2\theta + \sin^2\theta = 1\):\[(\cos^3\theta - 3 \cos\theta \sin^2\theta) + i (3 \cos^2\theta \sin\theta - \sin^3\theta)\]Simplify \(\cos\theta\) terms:\[= \cos^3\theta - 3 \cos\theta(1-\cos^2\theta) = 4\cos^3\theta - 3\cos\theta\]

Step 4: Equate Real Parts to Prove the Formula

The real part of the expression above corresponds to \(\cos(3\theta)\). Using our expansion and simplification:\[\cos(3\theta) = 4\cos^3\theta - 3\cos\theta\]This correctly matches the original equation to be proved using De Moivre's Theorem.

Key Concepts

Polar Form of Complex Numbers

The polar form of a complex number is a way of expressing it using the magnitude and angle relative to the positive x-axis. Instead of saying a complex number is in the form \(a + bi\), we use \(r(\cos \theta + i \sin \theta)\), where \(r\) is the modulus and \(\theta\) is the argument of the complex number.
This method is particularly useful in advanced calculus and other mathematical fields, as it simplifies the multiplication and powers of complex numbers. \(r\) is found using \( r = \sqrt{a^2 + b^2} \) and \(\theta\) is calculated using \ tan^{-1}(b/a) \.

• Modulus (r): The distance from the origin on the complex plane.
• Argument (\(\theta\)): The angle formed with the positive real axis.

De Moivre's Theorem leverages the polar form, making the calculation of powers and roots more manageable by manipulating the angle \(\theta\). This polar representation is not only crucial for calculations but also provides a geometric view of complex numbers.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions that hold true for any value of the variable. These identities are essential when working with De Moivre's Theorem and complex numbers in polar form.
Understanding these identities allows simplification and transformation of expressions, which is exactly what we need in problems involving powers like in De Moivre's Theorem:

• Basic Identities: \(\sin^2 \theta + \cos^2 \theta = 1\) is a vital identity that helps simplify expressions like \(\cos^2\theta\) and \(\sin^2\theta\).
• Pythagorean Identities: Derived from the unit circle, they relate the squares of sine and cosine to one.
• Angle Addition Formulas: Like \(\cos(\alpha + \beta)\) and \(\sin(\alpha + \beta)\) are used for expressing compound angles.

In the exercise problem, these identities help us rewrite and further simplify the expanded expressions, making the calculation of powers more straightforward.

Binomial Theorem

The Binomial Theorem provides a formula for expanding expressions raised to any power, such as \( (x + y)^n \). In the context of complex numbers and De Moivre's Theorem, this theorem helps expand expressions like \( (\cos \theta + i \sin \theta)^3 \).

• It states that every positive integer power can be expanded as a sum of terms of the form \( \binom{n}{k} x^{n-k} y^k \).
• Each term represents different combinations of multiplications of \(x\) and \(y\), involving binomial coefficients \( \binom{n}{k} \), which count the number of ways to choose \(k\) elements from \(n\) elements.

In calculus, we use it to expand expressions like \( (\cos \theta + i \sin \theta)^n \) into separate imaginary and real components, making simplification through identities easier. This approach lays the groundwork for finding powers and connecting trigonometric identities.

Complex Numbers in Calculus

Complex numbers play a significant role in calculus, particularly when dealing with integrals, derivatives, and series. They provide a comprehensive way of solving equations that real numbers can't, especially because of their ability to incorporate imaginary units.

• Complex Integration: Integrals in the complex plane can often be more straightforward due to symmetry and other analytical properties.
• Complex Differentiation: The process that contains rules like those in real calculus but applied to functions of a complex variable.
• Series and Convergence: Power series with complex numbers can describe functions over spherical surfaces rather than lines.

Understanding how complex numbers work in mathematics provides tools that simplify problems, particularly through the polar form and De Moivre’s Theorem. They highlight the deep connection between algebra, geometry, and calculus, allowing complex operations to be solved in a structured, meaningful way.