Problem 58 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The x-intercepts of the graph of the function \(y=2 \cos \theta+1\) are given by \(\theta = 2\pi n ± \pi/3\), for all integers \(n\).

Step 1: Understand the function

The given function is \(y=2 \cos \theta+1\). We are asked to find the x-intercepts of this function. In other words, we are looking for the values of \(\theta\) that make \(y = 0\).

Step 2: Set the function equal to zero

To find the x-intercepts, we need to set the function equal to zero and solve for \(\theta\). Therefore, we set \(y=0\) to obtain: \[0 = 2 \cos \theta + 1.\]

Step 3: Solve the equation obtained in Step 2

To solve for \(\theta\), we first subtract 1 from both sides of the equation: \[-1 = 2 \cos \theta.\] Next, we divide through by 2: \[-0.5 = \cos \theta.\] Finally, we use the inverse cosine function to solve for \(\theta\).

Step 4: Find the general solutions

The general solutions for the equation \(\cos \theta = -0.5\) are given by \(\theta = 2\pi n ± \pi/3\), where \(n\) is an integer. All these values of \(\theta\) will make \(y = 0\), and so they are all x-intercepts of the graph of the function.

Key Concepts

Understanding x-intercepts

X-intercepts are key points where a graph crosses the x-axis. They are crucial in analyzing functions and graphs.
For a function expressed as \(y = f(x)\), the x-intercepts occur where the output, \(y\), equals zero.
This involves solving the equation \(f(x) = 0\). For the given trigonometric function \(y = 2 \cos \theta + 1\), finding the x-intercepts means determining the values of \(\theta\) for which the function equals zero.

• Understand that x-intercepts are the roots or solutions of the equation, where the graph touches or crosses the x-axis.
• In trigonometric functions, these points depend on the periodic nature and transformations, such as shifts and scaling.

This makes x-intercepts a fundamental component in understanding and sketching graphs.

Inverse cosine and solving equations

The inverse cosine function, also known as arccosine, helps in finding angles corresponding to given cosine values.
When you encounter an equation like \(\cos \theta = -0.5\), the inverse cosine allows us to compute the angle \(\theta\).
Since cosine is periodic, its inverse will produce multiple angles that satisfy the initial equation.

• Inverse cosine is critical when solving trigonometric equations and provides principal values. Here, these values are often adjusted to find all possible solutions.
• The general solution in trigonometry considers the periodicity of the cosine function. For example, the solutions for \(\cos \theta = -0.5\) include specific principal angles and their periodic repeats.

Thus, using inverse cosine is vital for obtaining all solutions in trigonometric contexts, especially in cyclic patterns.

Exploring the cosine function

The cosine function is one of the fundamental trigonometric functions in mathematics.
It is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) interval.
The cosine of angle \(\theta\), expressed as \(\cos \theta\), varies between -1 and 1 throughout its cycle. This cyclic behavior is instrumental in solving equations with periodic solutions.

• The cosine graph is a wave-like structure that repeats over intervals of \(2\pi\). This periodic nature means multiple solutions for equations involving cosine, especially in the context of finding x-intercepts or roots.
• Transformations, such as multiplying by a constant or adding a number, shift the graph vertically or change its amplitude, respectively. In the function \(y = 2\cos \theta + 1\), this affects the graph's amplitude and vertical shift.

Understanding these characteristics and their transformations help demonstrate how the graph behaves and intersects the x-axis.