Problem 90 - Precalculus

Question

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

Step-by-Step Solution

Answer

By applying De Moivre's formula and comparing the imaginary parts, we confirm that \( \sin(2\theta) = 2\sin\theta\cos\theta \).

Step 1: Conversion to Polar Form

To use De Moivre's formula, start by representing the complex number in polar form. For a complex number with an angle \( \theta \), it is represented as \( \cos \theta + i \sin \theta \).

Step 2: Apply De Moivre's Formula

According to De Moivre's formula, for any real number \( \theta \) and integer \( n \), \( ( \cos \theta + i \sin \theta )^n = \cos(n\theta) + i \sin(n\theta) \). For \( n = 2 \), it gives \( ( \cos \theta + i \sin \theta )^2 = \cos(2\theta) + i \sin(2\theta) \).

Step 3: Expand Using Binomial Theorem

Expand \( ( \cos \theta + i \sin \theta )^2 \) using the binomial theorem: \[ ( \cos \theta + i \sin \theta )^2 = \cos^2 \theta + 2i \cos \theta \sin \theta - \sin^2 \theta \] This uses \( i^2 = -1 \).

Step 4: Separate Real and Imaginary Parts

The expanded form gives \( \cos^2 \theta - \sin^2 \theta + 2i \cos \theta \sin \theta \). The real part is \( \cos^2 \theta - \sin^2 \theta \) and the imaginary part is \( 2 \cos \theta \sin \theta \).

Step 5: Equate Imaginary Parts

From the equation \( \cos(2\theta) + i \sin(2\theta) = \cos^2 \theta - \sin^2 \theta + 2i \cos \theta \sin \theta \), consider the imaginary parts: \[ \sin(2\theta) = 2 \cos \theta \sin \theta \] This confirms the identity \( \sin(2\theta) = 2\sin\theta\cos\theta \).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are commonly expressed in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part, with \(i\) representing the imaginary unit such that \(i^2 = -1\).

• The real part is a standard number, like those we use every day.
• The imaginary part involves \(i\), which is essentially the square root of \(-1\).
• Complex numbers can be visualized on a plane called the complex plane, where the x-axis represents the real part, and the y-axis represents the imaginary part.

Understanding complex numbers is crucial because they allow solutions to equations that do not have real solutions, like \(x^2 + 1 = 0\). They also provide a powerful framework for solving a range of mathematical problems, particularly when utilizing them in polar forms.

Polar Form

The polar form of a complex number provides a different way to express the number by using a magnitude and an angle. This is particularly valuable when we are dealing with operations such as multiplication and division of complex numbers.

• The polar form is expressed as \(r(\cos\theta + i\sin\theta)\), where \(r\) is the magnitude (or modulus) and \(\theta\) is the angle (or argument).
• The magnitude \(r\) is calculated using the formula \(r = \sqrt{a^2 + b^2}\).
• The angle \(\theta\) is found using \(\theta = \tan^{-1}(\frac{b}{a})\), but one must be careful about the correct quadrant of \(\theta\).

This form is leveraged in De Moivre's Theorem to simplify exponentiation of complex numbers, as it allows them to be raised to powers or roots very efficiently.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables. These identities play a critical role in simplifying expressions and solving trigonometric equations.

• One of the key identities is the double angle identity, like \(\sin(2\theta) = 2 \sin\theta \cos\theta\).
• Such identities can often be derived using geometric methods or algebraic ones involving complex numbers, as in this exercise.
• Knowing and applying these identities can make complex mathematical problems easier to work with, especially in calculus and trigonometry.

Understanding these identities means you can often find shortcuts and elegant solutions to problems involving angles and oscillations, which is powerful in fields like physics and engineering.

Binomial Theorem

The binomial theorem provides a way to expand expressions that are raised to a power, specifically expressions in the form \((x + y)^n\). It is a cornerstone of algebra and has applications in calculus, probability, and other fields of mathematics.

• The theorem is expressed as:\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]
• Here, \(\binom{n}{k}\) represents the binomial coefficients, which can be calculated as \(\frac{n!}{k!(n-k)!}\).
• This expansion allows for complex expressions to be broken down into simpler parts.

In the context of this exercise, the binomial theorem is used to expand \((\cos \theta + i \sin \theta)^2\), which helps in equating the real and imaginary components to find trigonometric identities.