The Function $f$ Is Given By $f(x) = 5 + 7 \cos X$. For What Value Of $x$ On The Interval $\pi \ \textless \ X \ \textless \ 2\pi$ Does $f(x) = 0$?A. $\arccos \left(-\frac{5}{7}\right$\]B.

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Introduction

In this article, we will explore the function $f(x) = 5 + 7 \cos x$ and find the value of $x$ on the interval $\pi \ \textless \ x \ \textless \ 2\pi$ where $f(x) = 0$. This problem involves the use of trigonometric functions and their properties to solve for the roots of the function.

The Function $f(x)$

The function $f(x)$ is given by $f(x) = 5 + 7 \cos x$. This function is a combination of a constant term and a cosine term. The cosine term is the key to finding the roots of the function.

Finding the Roots of $f(x)$

To find the roots of $f(x)$, we need to set the function equal to zero and solve for $x$. This gives us the equation:

$$5 + 7 \cos x = 0$$

We can rearrange this equation to isolate the cosine term:

$$7 \cos x = -5$$

Now, we can divide both sides by 7 to get:

$$\cos x = -\frac{5}{7}$$

Using the Inverse Cosine Function

To find the value of $x$ that satisfies the equation $\cos x = -\frac{5}{7}$, we can use the inverse cosine function, denoted by $\arccos$. The inverse cosine function returns the angle whose cosine is a given value.

Using the inverse cosine function, we get:

$$x = \arccos \left(-\frac{5}{7}\right)$$

The Interval $\pi \ \textless \ x \ \textless \ 2\pi$

The problem states that we need to find the value of $x$ on the interval $\pi \ \textless \ x \ \textless \ 2\pi$. This means that the value of $x$ must be greater than $\pi$ and less than $2\pi$.

The Final Answer

Based on the previous calculations, we have found that the value of $x$ that satisfies the equation $\cos x = -\frac{5}{7}$ is:

$$x = \arccos \left(-\frac{5}{7}\right)$$

This value of $x$ is on the interval $\pi \ \textless \ x \ \textless \ 2\pi$.

Conclusion

In this article, we have explored the function $f(x) = 5 + 7 \cos x$ and found the value of $x$ on the interval $\pi \ \textless \ x \ \textless \ 2\pi$ where $f(x) = 0$. The value of $x$ is given by $x = \arccos \left(-\frac{5}{7}\right)$. This problem involves the use of trigonometric functions and their properties to solve for the roots of the function.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak

Discussion

This problem is a classic example of using trigonometric functions to solve for the roots of a function. The use of the inverse cosine function is a key step in solving this problem. The interval $\pi \ \textless \ x \ \textless \ 2\pi$ is a common interval used in trigonometric problems.

Related Problems

• Find the value of $x$ on the interval $0 \ \textless \ x \ \textless \ \pi$ where $f(x) = 0$.
• Find the value of $x$ on the interval $2\pi \ \textless \ x \ \textless \ 3\pi$ where $f(x) = 0$.

Solutions

• The value of $x$ on the interval $0 \ \textless \ x \ \textless \ \pi$ where $f(x) = 0$ is given by $x = \arccos \left(\frac{5}{7}\right)$.
• The value of $x$ on the interval $2\pi \ \textless \ x \ \textless \ 3\pi$ where $f(x) = 0$ is given by $x = \arccos \left(-\frac{5}{7}\right) + 2\pi$.
  Q&A: The Function $f$ and Its Roots =====================================

Q: What is the function $f(x)$?

A: The function $f(x)$ is given by $f(x) = 5 + 7 \cos x$. This function is a combination of a constant term and a cosine term.

Q: What is the interval $\pi \ \textless \ x \ \textless \ 2\pi$?

A: The interval $\pi \ \textless \ x \ \textless \ 2\pi$ is a common interval used in trigonometric problems. It represents the values of $x$ that are greater than $\pi$ and less than $2\pi$.

Q: How do we find the roots of $f(x)$?

A: To find the roots of $f(x)$, we need to set the function equal to zero and solve for $x$. This gives us the equation $5 + 7 \cos x = 0$. We can then rearrange this equation to isolate the cosine term and use the inverse cosine function to find the value of $x$.

Q: What is the inverse cosine function?

A: The inverse cosine function, denoted by $\arccos$, returns the angle whose cosine is a given value. In this case, we use the inverse cosine function to find the value of $x$ that satisfies the equation $\cos x = -\frac{5}{7}$.

Q: What is the value of $x$ on the interval $\pi \ \textless \ x \ \textless \ 2\pi$ where $f(x) = 0$?

A: The value of $x$ on the interval $\pi \ \textless \ x \ \textless \ 2\pi$ where $f(x) = 0$ is given by $x = \arccos \left(-\frac{5}{7}\right)$.

Q: What are some related problems?

A: Some related problems include finding the value of $x$ on the interval $0 \ \textless \ x \ \textless \ \pi$ where $f(x) = 0$ and finding the value of $x$ on the interval $2\pi \ \textless \ x \ \textless \ 3\pi$ where $f(x) = 0$.

Q: What are the solutions to these related problems?

A: The value of $x$ on the interval $0 \ \textless \ x \ \textless \ \pi$ where $f(x) = 0$ is given by $x = \arccos \left(\frac{5}{7}\right)$. The value of $x$ on the interval $2\pi \ \textless \ x \ \textless \ 3\pi$ where $f(x) = 0$ is given by $x = \arccos \left(-\frac{5}{7}\right) + 2\pi$.

Q: What is the significance of the interval $\pi \ \textless \ x \ \textless \ 2\pi$?

A: The interval $\pi \ \textless \ x \ \textless \ 2\pi$ is a common interval used in trigonometric problems. It represents the values of $x$ that are greater than $\pi$ and less than $2\pi$.

Q: What is the relationship between the function $f(x)$ and the cosine function?

A: The function $f(x)$ is a combination of a constant term and a cosine term. The cosine term is the key to finding the roots of the function.

Q: How do we use the inverse cosine function to find the roots of $f(x)$?

A: We use the inverse cosine function to find the value of $x$ that satisfies the equation $\cos x = -\frac{5}{7}$. This gives us the value of $x$ on the interval $\pi \ \textless \ x \ \textless \ 2\pi$ where $f(x) = 0$.

Q: What are some common intervals used in trigonometric problems?

A: Some common intervals used in trigonometric problems include the interval $\pi \ \textless \ x \ \textless \ 2\pi$ and the interval $0 \ \textless \ x \ \textless \ \pi$.