Problem 77 - Precalculus

Question

Use De Moivre's theorem to prove the identity \(\cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta\).

Step-by-Step Solution

Answer

Using De Moivre's Theorem and equating real parts, we prove \(\cos 2\theta = \cos^2\theta - \sin^2\theta\).

Step 1: Recall De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form, \(z = r \text{cis} \theta\), the \(n\)-th power can be expressed as \(z^n = r^n (\cos(n\theta) + i\sin(n\theta))\). For this exercise, we need to apply this theorem for \(n = 2\).

Step 2: Apply De Moivre's Theorem

Set \(z = \cos \theta + i \sin \theta\). According to De Moivre's Theorem, \(z^2 = (\cos \theta + i \sin \theta)^2 = \cos(2\theta) + i \sin(2\theta)\).

Step 3: Expand the Square Expression

Expand the expression \( (\cos \theta + i \sin \theta)^2\) using the identity \( (a+b)^2 = a^2 + 2ab + b^2\): \( \cos^2 \theta + 2i \cos \theta \sin \theta + i^2 \sin^2 \theta \). Since \(i^2 = -1\), this simplifies to \( \cos^2 \theta + 2i \cos \theta \sin \theta - \sin^2 \theta \).

Step 4: Equate Real Parts

From Step 2, \(z^2 = \cos(2\theta) + i \sin(2\theta)\) and the expanded form is \( \cos^2 \theta - \sin^2 \theta + 2i \cos \theta \sin \theta \). Equate the real parts of these expressions: \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \). This is the required identity.

Key Concepts

Complex Numbers

Complex numbers are fascinating because they extend our understanding of quantities beyond the real number line. A complex number is of the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).
Complex numbers can be visualized in a plane with the horizontal axis representing the real part \( a \) and the vertical axis the imaginary part \( b \).

• The combination of the real and imaginary components gives a point in this complex plane.
• The beauty of complex numbers lies in their ability to model two-dimensional phenomena.

When using De Moivre's Theorem, we look at complex numbers in a polar form, which simplifies operations like exponentiation to help solve trigonometric identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are pivotal in simplifying trigonometric expressions and proving relationships, like the one we see in this exercise.
The identity \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) is a classic example. It derives from angle sum identities and is useful for reducing the power of trigonometric functions.

• The double angle identity for cosine helps express angles twice as large in terms of functions of the angles themselves.
• This is crucial in calculus and physics, where angles often need to be simplified or rearranged.

Polar Form

Polar form is an alternative way of expressing complex numbers. Instead of using the real and imaginary parts, we represent them using a modulus \( r \) and an angle \( \theta \).
The polar form of a complex number \( z = r(\cos \theta + i \sin \theta) \) elegantly combines magnitude and direction and is sometimes expressed as \( z = r \text{cis} \theta \).

• This representation is valuable because multiplying complex numbers in polar form is easier than in rectangular form.
• It directly relates complex numbers to trigonometric functions, which is beneficial for solving problems using De Moivre's Theorem.

De Moivre's Theorem is often applied in this form, making powers and roots of complex numbers straightforward tasks.

Mathematical Proofs

Mathematical proofs are logical arguments that verify the truth of mathematical statements. They follow a structured process of evidence and reasoning.
The proof of \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) using De Moivre's Theorem exemplifies this rigor.

• First, we write the complex number in polar form and apply the theorem to square it.
• Then, we expand and simplify the expressions using algebra and trigonometric properties.
• Finally, by equating real parts, we verify the identity.

Throughout the proof, every step is reasoned out, ensuring the solution is precise and robust. Proofs not only confirm the validity of a statement but also enhance understanding of the underlying principles.