Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

\text { If } x \text { is a solution to } \cos x \sin x \cos (2 x) \sin (2 x)=0, \text { find } a l l \text { possible values of } \cos (x)

Step-by-Step Solution

Answer

Possible values of \( \cos(x) \) are any real number in the range [-1, 1].

Step 1: Applying the Null-Factor Property

The null-factor law says that if a product of factors equals zero, then at least one of the factors must be zero. This implies that either \( \cos(x) = 0 \), \( \sin(x) = 0 \), \( \cos(2x) = 0 \), or \( \sin(2x) = 0 \).

Step 2: Solving for x When Each Factor is Zero

Each factor yields possible values of x. For \( \cos(x) = 0 \), \( x = \frac{\pi}{2} + n\pi \) for \( n \) being any integer. For \( \sin(x) = 0 \), \( x = n\pi \). For \( \cos(2x) = 0 \), \( x = \frac{\pi}{4} + n\frac{\pi}{2} \). And for \( \sin(2x) = 0 \), \( x = n\frac{\pi}{2} \).

Step 3: Finding All Possible Values of cos(x)

Substituting all the values of \( x \), from the above step, into \( \cos(x) \) gives all possible values of \( \cos(x) \). For \( x = \frac{\pi}{2} + n\pi \), \( \cos(x) = 0 \). For \( x = n\pi \), \( \cos(x) = (-1)^n \). For \( x = \frac{\pi}{4} + n\frac{\pi}{2} \), \( \cos(x) = \frac{\sqrt{2}}{2}(-1)^n \). And for \( x = n\frac{\pi}{2} \), \( \cos(x) = (-1)^{n + 1} \). Thus, \( \cos(x) \) can be any value in the range of [-1, 1].

Key Concepts

Null-Factor Property

In mathematical equations, the null-factor property is invaluable for simplifying problems, especially when dealing with products equaling zero. The rule is simple: if the product of multiple factors is zero, then at least one of those factors must be zero. This is a handy principle when you're faced with equations like \( \cos x \sin x \cos(2x) \sin(2x) = 0 \). Here, we need only find one factor that is zero to progress in solving for \( x \).

• It allows breaking down complex equations into simpler ones.
• It helps in solving each smaller equation individually.

This makes the null-factor property a powerful starting point in mathematical problem-solving.

Trigonometric Identities

Trigonometric identities are crucial tools in solving trigonometric equations. They provide relationships between trigonometric functions that can simplify equations and unveil solutions. For instance, knowing that \[ \cos^2(x) + \sin^2(x) = 1 \] is key for understanding trigonometric equations.Additionally, double angle identities like \( \cos(2x) = 2\cos^2(x) - 1 \) or \( \sin(2x) = 2\sin(x)\cos(x) \) can transform and reduce complex expressions.

• They simplify calculations and help in expressing trigonometric terms differently.
• They enable the resolution of elaborate problems by breaking them into recognizable forms.

Familiarity with these identities is essential for efficient problem-solving in trigonometry.

Cosine Values

When working with trigonometric functions, cosine values hold specific importance, particularly due to their corresponding relationships with angles. These values can vary between -1 and 1 and are periodic, repeating every \( 2\pi \). In the context of the given exercise, understanding the specific values cosine can take for various angles \( x \) is critical. When analyzing solutions:

• For \( x = \frac{\pi}{2} + n\pi \), \( \cos(x) = 0 \).
• For integer values \( n \), \( x = n\pi \) implies \( \cos(x) = (-1)^n \).
• For \( x = \frac{\pi}{4} + n\frac{\pi}{2} \), \( \cos(x) = \frac{\sqrt{2}}{2}(-1)^n \).

Knowing these specific results helps ground the problem-solving process by identifying viable cosine outputs.

Mathematical Problem Solving

Approaching a mathematical problem effectively requires a systematic method, often called a problem-solving strategy. In this exercise, applying the null-factor property initially reduces the problem scope by identifying zero factors, which in turn simplifies equations. Once simplified, solve for each variable accordingly. Substitute them into the trigonometric function to find all possible solutions. This approach streamlines the process:

• Begin with examining and simplifying the given equation.
• Utilize trigonometric identities and properties to assist solution paths.
• Don't forget to check and interpret each solution for validity in context.

Problem-solving in math is not just about getting the answer, but understanding the process that leads to it, ensuring full comprehension and skill improvement.