Problem 18 - Precalculus with Calculus Previews

Question

Find the \(x\) -intercepts of the graph of the given function on the interval \([\mathrm{o}, 2 \pi]\). Then find all intercepts using periodicity. $$ y=1-2 \cos x $$

Step-by-Step Solution

Answer

The \(x\)-intercepts within \([0, 2\pi]\) are \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\). All intercepts are \(x = \frac{\pi}{3} + 2k\pi\) and \(x = \frac{5\pi}{3} + 2k\pi\), \(k\in\mathbb{Z}\).

Step 1: Understand the Objective

The goal is to find the \(x\)-intercepts of the function \(y = 1 - 2 \cos x\). An \(x\)-intercept occurs where the graph of the function crosses the \(x\)-axis, which is where \(y = 0\). We will solve for \(x\) within the interval \([0, 2\pi]\) and then use periodicity to find all solutions.

Step 2: Set the Equation to Zero

Set the function equal to zero to find the \(x\)-intercepts.\[1 - 2\cos x = 0\]

Step 3: Solve for \(\cos x\)

Rearrange the equation to solve for \(\cos x\):\[2\cos x = 1 \rightarrow \cos x = \frac{1}{2}\]

Step 4: Find the General Solutions

The \(\cos x = \frac{1}{2}\) occurs at specific angles within one period of cosine. These angles are:\[x = \frac{\pi}{3}, \quad x = \frac{5\pi}{3}\] because cosine is positive in the first and fourth quadrants within \([0, 2\pi]\).

Step 5: Use Periodicity to Find All Solutions

Cosine has a period of \(2\pi\), so the solutions repeat every \(2\pi\). The general solutions are:\[x = \frac{\pi}{3} + 2k\pi, \quad x = \frac{5\pi}{3} + 2k\pi\] where \(k\) is any integer.

Key Concepts

x-intercepts

In a function, the \(x\)-intercepts are where the function crosses the \(x\)-axis. This means the function value (\(y\)) is zero at these points. For the given function \(y = 1 - 2 \cos x\), to find the \(x\)-intercepts, you set \(y = 0\). This leads to:

• \(1 - 2 \cos x = 0\)

When you solve \(1 - 2 \cos x = 0\), you find \(\cos x = \frac{1}{2}\). This is key because it represents where the cosine value equals \(\frac{1}{2}\), indicating the angles where the graph crosses the \(x\)-axis. By solving \(\cos x = \frac{1}{2}\), you'll determine the angles within one cycle (the interval \([0, 2\pi]\)) where this occurs. This is essential for pinpointing the \(x\)-intercepts, showing how periodic trigonometric properties map the intersection points.

periodicity

In trigonometry, periodicity refers to the repeating nature of trigonometric functions over consistent intervals. For cosine, this period is \(2\pi\), meaning its values repeat every \(2\pi\) units along the \(x\)-axis. Understanding periodicity helps find all solutions to trigonometric equations.
To continue with our example, once we know \(\cos x = \frac{1}{2}\) is satisfied at \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\) within one cycle, periodicity tells us that these solutions repeat every \(2\pi\).

• The general solutions are \(x = \frac{\pi}{3} + 2k\pi\) and \(x = \frac{5\pi}{3} + 2k\pi\)

where \(k\) is any integer, representing the number of complete cycles added to each intercept, demonstrating how these points continuously repeat across the number line.

cosine function

The cosine function, a primary trigonometric function, describes a wave-like pattern, often used to model oscillatory phenomena. It's defined as the ratio of the adjacent side to the hypotenuse in a right triangle, or through the unit circle as the \(x\)-coordinate of a point on the circle.
In the equation \(y = 1 - 2 \cos x\), the cosine function is scaled and shifted:

• The amplitude is adjusted (scaled by 2), flipping it by multiplying with \(-2\).
• The vertical shift moves the cosine graph 1 unit upwards.

These transformations modify the basic cosine shape to fit specific function behavior. Recognizing these shifts and scalings is crucial for graphing and analyzing trigonometric functions. The cosine function's mathematical consistency and predictable nature due to its periodicity make it an essential component in various mathematical and real-world scenarios.