Problem 42 - Precalculus: Functions and Graphs

Question

Graph \(f\) on the interval \([-\pi, \pi] .\) (a) Estimate the \(x\) -intercepts. (b) Use sum-to-product formulas to find the exact values of the \(x\) -intercepts. $$f(x)=\cos 3 x-\cos 2 x$$

Step-by-Step Solution

Answer

The exact x-intercepts are \( x = -\frac{4\pi}{5}, 0, \frac{4\pi}{5} \).

Step 1: Understand the Function

The function given is \( f(x) = \cos 3x - \cos 2x \). We need to analyze this function on the interval \([-\pi, \pi]\). It's a trigonometric function involving cosine, and we are interested in finding where it crosses the x-axis, i.e., where \( f(x) = 0 \).

Step 2: Plot the Function

Graph the function \( f(x) \) over the interval \([-\pi, \pi]\). You can use graphing software or sketch it by understanding key points and behaviors of trigonometric functions. Look especially for the points where the graph crosses the x-axis, as these indicate the x-intercepts.

Step 3: Estimate the X-Intercepts from the Graph

By observing the graph of \( f(x) \), estimate the values of \( x \) in radians where the graph crosses the x-axis. Typically, these points should occur where the values of \( x \) are simple fractions of \( \pi \), such as \(-\frac{\pi}{2}, 0, \frac{\pi}{2} \), etc.

Step 4: Apply Sum-to-Product Formulas

Use the sum-to-product formula for cosines, which states that \( \cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \). Applying this to \( \cos 3x - \cos 2x \), we get: \[ f(x) = \cos 3x - \cos 2x = -2 \sin \left( \frac{5x}{2} \right) \sin \left( \frac{x}{2} \right) \]Now, set the equation to zero: \[ -2 \sin \left( \frac{5x}{2} \right) \sin \left( \frac{x}{2} \right) = 0. \]

Step 5: Solve for Exact X-Intercepts

The equation \(-2 \sin \left( \frac{5x}{2} \right) \sin \left( \frac{x}{2} \right) = 0\) implies that at least one of the sine factors must be zero. Thus, solve \[ \sin \left( \frac{5x}{2} \right) = 0 \] and \[ \sin \left( \frac{x}{2} \right) = 0. \]For \( \sin \left( \frac{5x}{2} \right) = 0 \), the solutions are given by:\[ \frac{5x}{2} = n\pi \Rightarrow x = \frac{2n\pi}{5}, \]where \( n \) is an integer.For \( \sin \left( \frac{x}{2} \right) = 0 \), the solutions are:\[ \frac{x}{2} = m\pi \Rightarrow x = 2m\pi, \]where \( m \) is an integer.

Step 6: Identify Valid Solutions Within Interval

Evaluate which solutions from step 5 fall within the interval \([-\pi, \pi]\): - For \( x = \frac{2n\pi}{5} \), check integers \( n \) such that \( -\pi \leq \frac{2n\pi}{5} \leq \pi \).- For \( x = 2m\pi \), within \([-\pi, \pi]\), the only valid value is \( x = 0 \).

Step 7: Conclude with Exact X-Intercepts

Conclude the solutions from Step 6:- From \( x = \frac{2n\pi}{5} \), valid values are \( x = -\frac{4\pi}{5}, 0, \frac{4\pi}{5} \) within the interval \([-\pi, \pi]\).- From \( x = 2m\pi \), only \( x = 0 \) is valid (and is already included). Thus, the exact x-intercepts are: \( x = -\frac{4\pi}{5}, 0, \frac{4\pi}{5} \).

Key Concepts

Sum-to-Product Formulas

The Sum-to-Product formulas are incredibly helpful when working with trigonometric functions. They allow us to transform sums or differences of trigonometric expressions into products. This transformation simplifies solving complex equations, especially when exploring x-intercepts.

For the function \( f(x) = \cos 3x - \cos 2x \), we use the specific sum-to-product formula for cosines:

• \( \cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \)

Applying this formula allows us to rewrite the function as:

• \( f(x) = -2 \sin \left( \frac{5x}{2} \right) \sin \left( \frac{x}{2} \right) \)

Breaking down the problem into factors helps in solving it efficiently. If either of the sine terms equals zero, the product is zero, revealing possible x-intercepts. Sum-to-product transforms a tricky equation into a much more manageable product format.

X-Intercepts

Finding x-intercepts involves solving the function for \( x \) where \( f(x) = 0 \). This means locating the points where the graph of the function crosses the x-axis. For trigonometric functions, this can often involve using specific angles or solving equations derived from symbolic transformations.

Once we apply the sum-to-product formula, we arrive at an expression in product form:

• \( -2 \sin \left( \frac{5x}{2} \right) \sin \left( \frac{x}{2} \right) = 0 \)

To solve this, set each sine factor to zero separately:

• \( \sin \left( \frac{5x}{2} \right) = 0 \)
• \( \sin \left( \frac{x}{2} \right) = 0 \)

These equations give solutions at specific points, calculated using multiples of \( \pi \). Considering the period of sine functions and the original interval \([-\pi, \pi]\), valid x-intercepts include points like \(-\frac{4\pi}{5}, 0, \frac{4\pi}{5}\). Recognizing repeat patterns and using key trigonometric identities streamlines the identification of these intercepts.

Graphing Trigonometric Functions

Graphing trigonometric functions can initially appear daunting, yet it becomes straightforward with practice and a solid understanding of key characteristics like amplitude, period, and phase shift.

For the function \( f(x) = \cos 3x - \cos 2x \), you'll first look for recognizable shapes and repeating patterns. Trigonometric graphs like cosine feature regular oscillations.
When graphing on \([-\pi, \pi]\), focus on:

• Identifying peaks, valleys, and axis crossings.
• Observing symmetry and periodicity, which are lifelines in trigonometric graphs.

Use tools, such as graphing calculators or software, to cross-verify your sketch. Watching how the graph behaves helps pinpoint x-intercepts visually before tackling them analytically. Moreover, understanding sums and differences of cosines assists in estimating these points accurately.
Grasping these attributes makes it easier to anticipate where the x-intercepts might fall, validating your calculations and offering insight into how trigonometric transformations influence the graph's path.