Problem 89 - Precalculus

Question

In advanced calculus, complex numbers in polar form are used extensively. Use De Moivre's formula to show that $$\cos (2 \theta)=\cos ^{2} \theta-\sin ^{2} \theta$$

Step-by-Step Solution

Answer

\( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \) using De Moivre's theorem.

Step 1: Express the Complex Number in Polar Form using De Moivre's Theorem

According to De Moivre's theorem, for any complex number expressed in polar form as \( z = \cos \theta + i \sin \theta \), raising it to the power \( n \) gives \( z^n = (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta) \). We want \( n = 2 \), so we will write: \((\cos \theta + i \sin \theta)^2 = \cos(2\theta) + i \sin(2\theta)\).

Step 2: Expand the Binomial Expression

Expand the left side of the equation: \((\cos \theta + i \sin \theta)^2\). Using the binomial theorem, this results in \((\cos^2 \theta + 2\cos \theta \cdot i \sin \theta + (i \sin \theta)^2)\).

Step 3: Simplify the Expression Using \(i^2 = -1\)

Recognize that \( i^2 = -1 \), so \((i \sin \theta)^2 = -\sin^2 \theta\). Thus, our expanded expression becomes \(\cos^2 \theta + 2i\cos \theta \sin \theta - \sin^2 \theta\).

Step 4: Separate Real and Imaginary Parts

The real part on the left side is \(\cos^2 \theta - \sin^2 \theta\) and the imaginary part is \(2\cos \theta \sin \theta\). By De Moivre's Theorem, these correspond to \(\cos(2\theta)\) and \(\sin(2\theta)\), respectively. We are interested in the real part, \(\cos^2 \theta - \sin^2 \theta\).

Step 5: Conclude with De Moivre's Formula

Since the real part of the expansion \(\cos^2 \theta - \sin^2 \theta\) matches \(\cos(2\theta)\) on the right side of De Moivre's theorem, we conclude that: \(\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\).

Key Concepts

Complex Numbers

Complex numbers form an essential part of mathematics, especially advanced calculus and engineering. They consist of a real part and an imaginary part, written as \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component, with \(i\) being the imaginary unit satisfying \(i^2 = -1\).
Originating in solving quadratic equations, complex numbers allow for the extension of real number operations. They enable the computation of solutions, which would otherwise not be possible due to negative square roots.
By representing periodic phenomena, they also play a significant role in fields like signal processing and quantum mechanics.
To make operations like multiplication and division more manageable, complex numbers can be expressed in the polar form, enhancing their utility in mathematical computations.

Polar Form

Transforming complex numbers into polar form facilitates many mathematical operations. Polar form represents a complex number as \( z = r(\cos \theta + i \sin \theta) \), where \(r\) is the modulus (distance from the origin in the complex plane), and \(\theta\) is the argument (angle from the positive x-axis).
This representation leverages the relationship between trigonometric functions and complex numbers, providing an insightful way to analyze and compute complex operations.
Using polar form is especially beneficial when employing De Moivre's Theorem, allowing for straightforward calculation of powers and roots of complex numbers. The theorem is succinctly expressed as: for a complex number \(z\), \([z^n = r^n(\cos(n\theta) + i\sin(n\theta))]\).
Here, target solutions become more accessible, making polar form a powerful tool in complex number analysis.

Trigonometric Identities

Trigonometric identities are formulas that involve the angles and sides of triangles. They are crucial for simplifying expressions and solving equations involving trigonometric functions.
One of these identities is the double angle identity for cosine: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \). This can be derived using De Moivre's theorem, as described in the solution.
These identities help in converting and simplifying complex trigonometric expressions, making them essential in both pure and applied mathematics.
They are especially handy in calculus and geometry, where they assist in solving integrals, derivatives, and proving the equivalence of different trigonometric expressions.

Binomial Theorem

The binomial theorem provides a way to expand expressions of the form \((x + y)^n\). It states:

• \((x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\)

where \(\binom{n}{k}\) is a binomial coefficient, which is a specific way of selecting elements from a set.
In the context of complex numbers and De Moivre's Theorem, the binomial theorem helps expand the expression \((\cos \theta + i \sin \theta)^n\), aiding in verifying identities and other complex calculations.
This powerful theorem streamlines polynomial expansions, enabling a systematic approach to handling powers, which is particularly useful in algebra and related fields.