Using the Properties of the Cosine Function

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In mathematics, the concept of x $}$ -intercepts is crucial in understanding the behavior of various functions, including trigonometric functions. An <span class="katex-error" title="ParseError KaTeX parse error: Expected '', got 'EOF' at end of input: " style="color#cc0000">{ x $$-intercept is the point at which the graph of a function crosses the x-axis, meaning the y-coordinate is zero. In this article, we will delve into the concept of x $}$ -intercepts in the context of the function <span class="katex-error" title="ParseError KaTeX parse error: Expected '', got 'EOF' at end of input: " style="color#cc0000">{ y = \cos \left(\frac{1{2} x\right) $}$ .

What are ${$ x $}$-Intercepts?

An ${$ x $}$-intercept is a point on the graph of a function where the y-coordinate is zero. In other words, it is the point at which the graph crosses the x-axis. To find the ${$ x $}$-intercept of a function, we need to set the function equal to zero and solve for x.

The Function ${$ y = \cos \left(\frac{1}{2} x\right) $}$

The given function is ${$ y = \cos \left(\frac{1}{2} x\right) $}$. This is a trigonometric function, specifically a cosine function, with a coefficient of ${$ \frac{1}{2} $}$ in front of the x variable. The cosine function has a period of ${$ 2 \pi $}$, which means it repeats every ${$ 2 \pi $}$ units of x.

Finding the ${$ x $}$-Intercept

To find the ${$ x $}$-intercept of the function ${$ y = \cos \left(\frac{1}{2} x\right) $}$, we need to set the function equal to zero and solve for x. This means we need to find the value of x for which ${$ \cos \left(\frac{1}{2} x\right) = 0 $}$.

Using the Properties of the Cosine Function

The cosine function has a value of zero at odd multiples of ${$ \frac{\pi}{2} $}$. This means that ${$ \cos \left(\frac{1}{2} x\right) = 0 $}$ when ${$ \frac{1}{2} x = \frac{\pi}{2} + k \pi $}$, where k is an integer.

Solving for x

To solve for x, we need to multiply both sides of the equation by 2, which gives us ${$ x = \pi + 2 k \pi $}$. This is the general solution for the ${$ x $}$-intercept of the function ${$ y = \cos \left(\frac{1}{2} x\right) $}$.

Evaluating the Options

Now that we have the general solution for the ${$ x $}$-intercept, we can evaluate the options given in the problem.

Option A: ${$ 0 $}$ is not a solution to the equation ${$ x = \pi + 2 k \pi $}$.
Option B: ${$ \frac{\pi}{2} $}$ is not a solution to the equation ${$ x = \pi + 2 k \pi $}$.
Option C: ${$ \pi $}$ is a solution to the equation ${$ x = \pi + 2 k \pi $}$ when k = 0.
Option D: ${$ 2 \pi $}$ is a solution to the equation ${$ x = \pi + 2 k \pi $}$ when k = 1.

Conclusion

In conclusion, the correct answer is ${$ \pi $}$, which is option C. This is because ${$ \pi $}$ is a solution to the equation ${$ x = \pi + 2 k \pi $}$ when k = 0.

Final Answer

In our previous article, we discussed the concept of ${$ x $}$-intercepts in the context of the function ${$ y = \cos \left(\frac{1}{2} x\right) $}$. We also provided a detailed solution to find the ${$ x $}$-intercept of the function. In this article, we will address some frequently asked questions related to ${$ x $}$-intercepts in trigonometric functions.

Q: What is the significance of ${$ x $}$-intercepts in trigonometric functions?

A: ${$ x $}$-intercepts are significant in trigonometric functions as they provide valuable information about the behavior of the function. The ${$ x $}$-intercept of a function is the point at which the graph of the function crosses the x-axis, meaning the y-coordinate is zero. This information can be used to determine the period, amplitude, and phase shift of the function.

Q: How do I find the ${$ x $}$-intercept of a trigonometric function?

A: To find the ${$ x $}$-intercept of a trigonometric function, you need to set the function equal to zero and solve for x. This involves using the properties of the trigonometric function, such as the period and amplitude, to determine the value of x.

Q: What are some common mistakes to avoid when finding ${$ x $}$-intercepts?

A: Some common mistakes to avoid when finding ${$ x $}$-intercepts include:

Not setting the function equal to zero before solving for x
Not using the correct properties of the trigonometric function
Not considering the period and amplitude of the function
Not checking for extraneous solutions

Q: Can you provide an example of finding the ${$ x $}$-intercept of a trigonometric function?

A: Let's consider the function ${$ y = \sin \left(\frac{1}{3} x\right) $}$. To find the ${$ x $}$-intercept of this function, we need to set the function equal to zero and solve for x.

${$ \sin \left(\frac{1}{3} x\right) = 0 $}$

Using the properties of the sine function, we know that ${$ \sin \left(\frac{1}{3} x\right) = 0 $}$ when ${$ \frac{1}{3} x = k \pi $}$, where k is an integer.

Solving for x, we get:

${$ x = 3 k \pi $}$

This is the general solution for the ${$ x $}$-intercept of the function ${$ y = \sin \left(\frac{1}{3} x\right) $}$.

Q: What is the difference between the ${$ x $}$-intercept and the ${$ y $}$-intercept?

A: The ${$ x $}$-intercept is the point at which the graph of a function crosses the x-axis, meaning the y-coordinate is zero. The ${$ y $}$-intercept, on the other hand, is the point at which the graph of a function crosses the y-axis, meaning the x-coordinate is zero.

Q: Can you provide some real-world applications of ${$ x $}$-intercepts in trigonometric functions?

A: Yes, ${$ x $}$-intercepts in trigonometric functions have several real-world applications, including:

Modeling periodic phenomena, such as sound waves and light waves
Analyzing the behavior of electrical circuits and electronic devices
Studying the motion of objects in physics and engineering
Modeling population growth and decline in biology and ecology

Conclusion

In conclusion, ${$ x $}$-intercepts in trigonometric functions are an essential concept in mathematics and have numerous real-world applications. By understanding how to find the ${$ x $}$-intercept of a trigonometric function, you can gain valuable insights into the behavior of the function and apply this knowledge to solve problems in various fields.

Final Answer

The final answer is: $\boxed{C}$