Problem 39 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\cos ^{2} 2 x-\sin ^{2} 2 x=0$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{(2n+1)\pi}{8} \), \( n \in \mathbb{Z} \).

Step 1: Apply the Double Angle Identity

Recognize that the given equation resembles a trigonometric identity. The expression \( \cos^2 \theta - \sin^2 \theta \) is equivalent to \( \cos(2\theta) \). Therefore, convert the equation \( \cos^2 (2x) - \sin^2 (2x) = 0 \) into \( \cos(4x) = 0 \).

Step 2: Solve the Simplified Trigonometric Equation

Now solve the equation \( \cos(4x) = 0 \). The cosine function is equal to zero at odd multiples of \( \frac{\pi}{2} \), so we have the equation \( 4x = (2n+1)\frac{\pi}{2} \), where \( n \) is an integer.

Step 3: Solve for x

To find \( x \), divide both sides of the equation from Step 2 by 4: \( x = \frac{(2n+1)\pi}{8} \).

Step 4: General Solution

Therefore, the general solution for the original equation is \( x = \frac{(2n+1)\pi}{8} \), where \( n \) is any integer.

Key Concepts

Double Angle Identity

The double angle identity is a useful trigonometric identity that allows us to simplify expressions involving angles that are twice a given angle. For example, the double angle identity for cosine is written as:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)
• It can also be expressed as \( \cos(2\theta) = 2\cos^2(\theta) - 1 \) or \( \cos(2\theta) = 1 - 2\sin^2(\theta) \).

These identities are derived from the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). In solving trigonometric equations, these identities help to convert complex expressions into simpler forms. By recognizing these forms, as in the given exercise, we can tackle more complex trigonometric problems effectively. This transformation is key in solving the given equation.

Cosine Function

The cosine function, represented by \( \cos(x) \), is one of the fundamental trigonometric functions. It describes the relationship between the angle and the length of the adjacent side of a right triangle divided by the hypotenuse.

• It is periodic, with a period of \( 2\pi \), meaning \( \cos(x + 2\pi) = \cos(x) \).
• The function is even, satisfying \( \cos(-x) = \cos(x) \).
• It takes on values between -1 and 1, inclusive.

In this problem, the cosine function is used in its doubled angle form \( \cos(4x) \). Understanding the properties of cosine, including its zeros, is essential. Cosine is zero at odd multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2}, \frac{3\pi}{2}, ... \). Knowing where cosine equals zero enables us to solve the simplified equation.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions that require solutions over specified intervals. Solving these equations often involves using identities like the double angle identity, algebraic manipulation, and understanding of trigonometric functions' core properties.
In the exercise, the trigonometric equation \( \cos^2(2x) - \sin^2(2x) = 0 \) leverages the double angle identity to become \( \cos(4x) = 0 \). Therefore, the challenge becomes finding values of \( x \) that satisfy this condition.

• To solve, recognize that cosine being zero corresponds to specific angles.
• Translate the trigonometric condition into a solvable algebraic form.

This method turns seemingly complex trigonometric expressions into solvable problems by understanding and applying key functions and identities effectively.

General Solution

Finding the general solution of a trigonometric equation involves identifying all possible solutions, often expressed in terms of integer products of a period.

• For the equation \( \cos(4x) = 0 \), we find its general solution by solving \( 4x = (2n+1)\frac{\pi}{2} \).
• Divide by 4 to solve for \( x \): \( x = \frac{(2n+1)\pi}{8} \).

The integer \( n \) represents different cycles of the function, accounting for its periodic nature. This general solution accounts for all possible solutions to the equation across its infinite domain, important in understanding how periodic functions behave. This approach allows us to chart all solutions beyond any singular interval, providing a comprehensive understanding of the solution space.