Problem 54 - Precalculus with Calculus Previews

Question

Find the first three \(x\) -intercepts of the graph of the given function on the positive \(x\) -axis. $$ f(x)=2 \cos \left(x+\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The first three x-intercepts are \( \frac{\pi}{4}, \frac{5\pi}{4}, \frac{9\pi}{4} \).

Step 1: Understanding X-Intercepts

The x-intercepts of a function occur where the function equals zero. Therefore, to find the x-intercepts of the function \( f(x) = 2 \cos \left(x + \frac{\pi}{4}\right) \), we need to solve the equation \( 2 \cos \left(x + \frac{\pi}{4}\right) = 0 \).

Step 2: Solving the Trigonometric Equation

Divide both sides by 2 to simplify, yielding \( \cos \left(x + \frac{\pi}{4}\right) = 0 \). The cosine function equals zero at odd multiples of \( \frac{\pi}{2} \), so we set \( x + \frac{\pi}{4} = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Step 3: Express x in Terms of n

To isolate \( x \), subtract \( \frac{\pi}{4} \) from both sides: \( x = \frac{\pi}{2} + n\pi - \frac{\pi}{4} \). Simplify to get \( x = \frac{2\pi}{4} + n\pi - \frac{\pi}{4} \) which simplifies further to \( x = \frac{\pi}{4} + n\pi \).

Step 4: Finding the First Three Positive X-Intercepts

To find the first three positive x-intercepts, we substitute successive integer values of \( n \) starting from zero. When \( n = 0 \), \( x = \frac{\pi}{4} \). For \( n = 1 \), \( x = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \). For \( n = 2 \), \( x = \frac{\pi}{4} + 2\pi = \frac{9\pi}{4} \). These are the first three positive x-intercepts.

Key Concepts

Trigonometric Equations

Trigonometric equations involve functions like sine and cosine, which are fundamental in understanding periodic phenomena. These functions repeat at regular intervals, making their graphs form predictable patterns. Solving trigonometric equations typically involves determining angles that satisfy specific conditions for these functions, such as where they equal a particular value.

• The process involves isolating the trigonometric function.
• Next, identify the standard angles that satisfy the equation, often informed by what values the trigonometric function takes on at common angles.
• Finally, you generalize the solution, accounting for the function's periodicity by introducing multiplication of a full period (e.g., \(2\pi\) for sine and cosine) added to the general angle.

For example, solving \(\cos(x) = 0\) involves recognizing that cosine is zero at \(x = \frac{\pi}{2}\) plus integer multiples of \(\pi\), yielding the general solution \[x = \frac{\pi}{2} + n\pi, \text{where } n \text{ is an integer.}\]By recognizing these properties, finding solutions becomes more straightforward and predictive.

Cosine Function

The cosine function is one of the primary trigonometric functions and is closely associated with the unit circle, where the angle \(x\) corresponds to the x-coordinate of a point on the circle. This function has several important properties:

• The cosine function evaluates to zero at odd multiples of \(\frac{\pi}{2}\).
• It reaches its maximum value of 1 at multiples of \(2\pi\), and a minimum value of -1 at odd multiples of \(\pi\).
• Cosine is periodic, repeating every \(2\pi\) radians.

This periodicity and predictability make it very useful in various applications, including physics and engineering. When solving an equation like \(\cos(x + \frac{\pi}{4}) = 0\), tapping into these properties allows us to find all solutions by evaluating the phase shift \(+\frac{\pi}{4}\), and adjusting for this by solving for \(x\) in our equation, leading us directly to the accurate x-intercepts.

Positive X-Axis

The positive x-axis refers to that portion of the coordinate plane where x-values are positive. It lies to the right of the origin along the horizontal line. In many equations, identifying the positive x-axis is critical since it determines specific constraints or limits to the solution set.For cosine equations, when asked to find x-intercepts on the positive x-axis, we are looking for the smallest positive \(x\) values that satisfy the trigonometric condition given. Many problems limit solutions to this range to focus on immediate, practical solutions rather than a general set that includes negative or expansive positive values derived through multiple full cycles of the function.

• This reduces the number of possible x-values to a more manageable set, aiding in real-world application.
• Additionally, focusing on the first few positive intercepts gives insight into future behavior of functions at periodic intervals.

Overall, looking at solutions on the positive x-axis helps in better understanding the immediate beginnings of function behaviors, useful particularly when visualizing graphs or predicting behaviors.