Problem 60 - Algebra 2

Question

Find the \(x\) -intercepts of the graph of each function. $$ y=\cos ^{2} \theta-1 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( \theta = 2n\pi \) and \( \theta = (2n + 1)\pi \), where \( n \) is an integer.

Step 1: Simplify the equation

The equation to simplify becomes: \( \cos^{2}\theta = 1 \)

Step 2: Take the square root of both sides

Then, considering that both the positive and negative square roots must be taken into account, the equation becomes: \(\cos\theta = \pm 1\).

Step 3: Solution for \( \theta \)

Now, remember that \( \cos\theta = 1 \) when \( \theta = 2n\pi \) and \( \cos\theta = -1 \) when \( \theta = (2n + 1)\pi \), where \( n \) is an integer. So our solution set becomes \( \theta = 2n\pi \) and \( \theta = (2n + 1)\pi \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, representing the x-coordinate of a point on the unit circle. When graphed, it creates a wave-like pattern that repeats every full cycle, or every \(2\pi\) radians.
The basic form of the cosine function is \(y = \cos\theta\), where \(\theta\) is an angle measured in radians. This function oscillates between 1 and -1, with key points at specific angles:

• \(\cos 0 = 1\)
• \(\cos \pi/2 = 0\)
• \(\cos \pi = -1\)
• \(\cos 3\pi/2 = 0\)
• \(\cos 2\pi = 1\)

The cosine function is even, meaning it is symmetric around the y-axis: \(\cos(-\theta) = \cos(\theta)\).
This property is particularly useful when solving trigonometric equations, as it helps identify solutions over various intervals.

Graph Intercepts

Graph intercepts are the points where a graph intersects the x-axis or y-axis. For trigonometric functions like cosine, these intercepts are key to understanding the function's behavior over a cycle.

X-Intercepts of the Cosine Function

To find the x-intercepts of a function like \(y = \cos^2 \theta - 1\), set the function equal to zero and solve for \(\theta\):
1. First, set \(\cos^2 \theta - 1 = 0\), resulting in \(\cos^2 \theta = 1\).
2. Taking the square root gives \(\cos\theta = \pm 1\).
3. Solve for \(\theta\) to find the x-intercepts. For \(\cos\theta = 1\), \(\theta = 2n\pi\) and for \(\cos\theta = -1\), \(\theta = (2n + 1)\pi\), where \(n\) is an integer.
Graph analysis shows these points where the graph crosses the x-axis.

Using Intercepts in Problem Solving

Intercepts help in sketching graphs quickly and figuring out intervals where the function changes sign, which is essential in solving equations graphically and analytically.

Trigonometric Equations

Trigonometric equations are mathematical statements that involve trigonometric functions—such as sine, cosine, and tangent—and are set equal to a value. Solving these equations often requires understanding the periodic and symmetrical properties of the trigonometric functions.

General Steps to Solve Trigonometric Equations

1. **Simplify the Equation**: Use identities and algebraic manipulations to simplify the equation. For example, converting \(\cos^2 \theta - 1 = 0\) to \(\cos^2 \theta = 1\).
2. **Apply the Unit Circle**: Recognize the angles where specific trigonometric values occur. This is essential for finding solutions over different intervals. For the cosine function, equate \(\cos \theta = \pm 1\) to solve for \(\theta\).
3. **Consider the Periodicity**: Due to their periodic nature, trigonometric solutions repeat every full cycle. Thus, include the general solution formula. For \(\cos \theta = 1\), it is \(\theta = 2n\pi\), and for \(\cos \theta = -1\), it is \(\theta = (2n + 1)\pi\), where \(n\) is an integer.
Understanding the behavior of trigonometric equations helps tackle more complex problems that include multiple steps and require diverse mathematical strategies.