Problem 14 - Algebra and Trigonometry

Question

Find all solutions of the equation. $$\sec x(2 \cos x-\sqrt{2})=0$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{\pi}{4} + 2n\pi \) and \( x = -\frac{\pi}{4} + 2n\pi \), \( n \in \mathbb{Z} \).

Step 1: Analyze the Equation

The equation given is \( \sec x (2 \cos x - \sqrt{2}) = 0 \). This equation is in the product form, meaning it can be split into two distinct equations based on the zero-product property.

Step 2: Solve \( \sec x = 0 \)

The secant of an angle, \( \sec x \), is undefined where \( \cos x = 0 \). This occurs when \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an odd integer. However, since secant is never zero, the equation \( \sec x = 0 \) has no solutions.

Step 3: Solve \( 2 \cos x - \sqrt{2} = 0 \)

Rearrange the equation to find \( \cos x \):\[ 2 \cos x = \sqrt{2} \]\[ \cos x = \frac{\sqrt{2}}{2} \].This occurs at angles where \( x = \frac{\pi}{4} + 2n\pi \) and \( x = -\frac{\pi}{4} + 2n\pi \), where \( n \) is any integer.

Step 4: Conclude the Solutions

Since \( \sec x = 0 \) provides no solutions, the solutions to the original equation are satisfied by the second part of the product equation, \( 2 \cos x - \sqrt{2} = 0 \). Thus, the solutions are \( x = \frac{\pi}{4} + 2n\pi \) and \( x = -\frac{\pi}{4} + 2n\pi \) for any integer \( n \).

Key Concepts

Zero-Product Property

The zero-product property is a useful tool when solving trigonometric equations. It states that if the product of two expressions equals zero, then at least one of the expressions must equal zero. In the equation \( \sec x (2 \cos x - \sqrt{2}) = 0 \), we apply this property.

Using the zero-product property, we split the original equation into two smaller equations:

• \( \sec x = 0 \)
• \( 2 \cos x - \sqrt{2} = 0 \)

Each part can be solved independently. This simplifies complex equations into more manageable ones.

For \( \sec x = 0 \), however, we discover that it has no solutions because secant is never zero. This step immediately reduces our solution space, letting us focus on the more promising equation, \( 2 \cos x - \sqrt{2} = 0 \). This property is particularly powerful because it helps break down the problem into solvable pieces.

Secant and Cosine Relationship

The secant function, \( \sec x \), is directly related to the cosine function. It is defined as \( \sec x = \frac{1}{\cos x} \).

This relationship means that the secant is undefined wherever cosine equals zero. However, crucially, \( \sec x \) being zero would imply \( \cos x = \infty \), which is impossible. That's why \( \sec x = 0 \) has no solutions.

Understanding the secant-cosine relationship helps us predict where issues with definition occur. Here, solving \( \sec x = 0 \) leads nowhere, guiding focus on \( \cos x \) values satisfying differing conditions. By exploring \( 2 \cos x - \sqrt{2} = 0 \), we find more information about solutions because it revolves around regular cosine values.

General Solutions for Trigonometric Equations

Finding general solutions for trigonometric equations is about identifying all possible solutions. Trigonometric functions like cosine replicate their cycle every \(2\pi\).

The equation \( 2 \cos x - \sqrt{2} = 0 \) simplifies to \( \cos x = \frac{\sqrt{2}}{2} \). This cosine value is associated with specific angle solutions:

• \( x = \frac{\pi}{4} + 2n\pi \)
• \( x = -\frac{\pi}{4} + 2n\pi \)

Here, \( n \) is any integer, representing the rotation number of the angle around the unit circle.

Providing solutions that incorporate \( n \) ensures that all cycles of the trigonometric function are accounted for, yielding comprehensive coverage of all angles that satisfy the equation.