What Are The Zeros Of The Function F ( X ) = 2 − 4 Cos ⁡ X F(x) = 2 - 4 \cos X F ( X ) = 2 − 4 Cos X ?A. X = Π 6 + 2 Π N X = \frac{\pi}{6} + 2\pi N X = 6 Π ​ + 2 Πn And X = 5 Π 6 + 2 Π N X = \frac{5\pi}{6} + 2\pi N X = 6 5 Π ​ + 2 Πn B. X = 7 Π 6 + 2 Π N X = \frac{7\pi}{6} + 2\pi N X = 6 7 Π ​ + 2 Πn And X = 11 Π 6 + 2 Π N X = \frac{11\pi}{6} + 2\pi N X = 6 11 Π ​ + 2 Πn C. $x =

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Understanding the Problem

The problem asks us to find the zeros of the function $f(x) = 2 - 4 \cos x$. In other words, we need to find the values of $x$ for which the function $f(x)$ equals zero. To solve this problem, we will use the properties of the cosine function and the concept of periodicity.

Recalling the Properties of the Cosine Function

The cosine function has a period of $2\pi$, which means that the graph of the cosine function repeats itself every $2\pi$ units. This means that if we know the value of the cosine function at a certain point, we can find the value of the cosine function at any other point by adding or subtracting multiples of $2\pi$.

Finding the Zeros of the Function

To find the zeros of the function $f(x) = 2 - 4 \cos x$, we need to set the function equal to zero and solve for $x$. This gives us the equation:

$$2 - 4 \cos x = 0$$

Simplifying this equation, we get:

$$-4 \cos x = -2$$

Dividing both sides by $-4$, we get:

$$\cos x = \frac{1}{2}$$

Recalling the Values of the Cosine Function

We know that the cosine function takes on the value of $\frac{1}{2}$ at certain points in its period. Specifically, the cosine function takes on the value of $\frac{1}{2}$ at the points $\frac{\pi}{6}$ and $\frac{5\pi}{6}$.

Finding the Zeros of the Function

Since the cosine function takes on the value of $\frac{1}{2}$ at the points $\frac{\pi}{6}$ and $\frac{5\pi}{6}$, we can conclude that the zeros of the function $f(x) = 2 - 4 \cos x$ are given by:

$$x = \frac{\pi}{6} + 2\pi n$$

and

$$x = \frac{5\pi}{6} + 2\pi n$$

where $n$ is an integer.

Conclusion

In conclusion, the zeros of the function $f(x) = 2 - 4 \cos x$ are given by:

$$x = \frac{\pi}{6} + 2\pi n$$

and

$$x = \frac{5\pi}{6} + 2\pi n$$

where $n$ is an integer.

Answer

The correct answer is:

A. $x = \frac{\pi}{6} + 2\pi n$ and $x = \frac{5\pi}{6} + 2\pi n$

Discussion

This problem is a classic example of how to use the properties of the cosine function to find the zeros of a trigonometric function. The key idea is to recall the values of the cosine function and use the concept of periodicity to find the zeros of the function.

Related Problems

• Find the zeros of the function $f(x) = 3 - 2 \sin x$.
• Find the zeros of the function $f(x) = 1 + 4 \sin x$.
• Find the zeros of the function $f(x) = 2 - 3 \cos x$.

Solutions

• The zeros of the function $f(x) = 3 - 2 \sin x$ are given by:

$$x = \frac{\pi}{2} + 2\pi n$$

and

$$x = \frac{3\pi}{2} + 2\pi n$$

where $n$ is an integer.

• The zeros of the function $f(x) = 1 + 4 \sin x$ are given by:

$$x = \frac{\pi}{2} + 2\pi n$$

and

$$x = \frac{3\pi}{2} + 2\pi n$$

where $n$ is an integer.

• The zeros of the function $f(x) = 2 - 3 \cos x$ are given by:

$$x = \frac{\pi}{3} + 2\pi n$$

and

$$x = \frac{5\pi}{3} + 2\pi n$$

where $n$ is an integer.

Conclusion

In conclusion, the zeros of the function $f(x) = 2 - 4 \cos x$ are given by:

$$x = \frac{\pi}{6} + 2\pi n$$

and

$$x = \frac{5\pi}{6} + 2\pi n$$

where $n$ is an integer.

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Q: What is the function $f(x) = 2 - 4 \cos x$?

A: The function $f(x) = 2 - 4 \cos x$ is a trigonometric function that involves the cosine function. The function takes on the value of $2 - 4 \cos x$ for any given value of $x$.

Q: What are the zeros of the function $f(x) = 2 - 4 \cos x$?

A: The zeros of the function $f(x) = 2 - 4 \cos x$ are the values of $x$ for which the function $f(x)$ equals zero. In other words, we need to find the values of $x$ for which $2 - 4 \cos x = 0$.

Q: How do we find the zeros of the function $f(x) = 2 - 4 \cos x$?

A: To find the zeros of the function $f(x) = 2 - 4 \cos x$, we need to set the function equal to zero and solve for $x$. This gives us the equation:

$$2 - 4 \cos x = 0$$

Simplifying this equation, we get:

$$-4 \cos x = -2$$

Dividing both sides by $-4$, we get:

$$\cos x = \frac{1}{2}$$

Q: What are the values of $x$ for which $\cos x = \frac{1}{2}$?

A: The values of $x$ for which $\cos x = \frac{1}{2}$ are given by:

$$x = \frac{\pi}{6} + 2\pi n$$

and

$$x = \frac{5\pi}{6} + 2\pi n$$

where $n$ is an integer.

Q: What is the significance of the values of $x$ for which $\cos x = \frac{1}{2}$?

A: The values of $x$ for which $\cos x = \frac{1}{2}$ are the zeros of the function $f(x) = 2 - 4 \cos x$. In other words, these values of $x$ make the function $f(x)$ equal to zero.

Q: How do we use the values of $x$ for which $\cos x = \frac{1}{2}$ to find the zeros of the function $f(x) = 2 - 4 \cos x$?

A: We use the values of $x$ for which $\cos x = \frac{1}{2}$ to find the zeros of the function $f(x) = 2 - 4 \cos x$ by substituting these values into the equation $2 - 4 \cos x = 0$. This gives us the zeros of the function $f(x) = 2 - 4 \cos x$.

Q: What are the zeros of the function $f(x) = 2 - 4 \cos x$?

A: The zeros of the function $f(x) = 2 - 4 \cos x$ are given by:

$$x = \frac{\pi}{6} + 2\pi n$$

and

$$x = \frac{5\pi}{6} + 2\pi n$$

where $n$ is an integer.

Q: What is the significance of the zeros of the function $f(x) = 2 - 4 \cos x$?

A: The zeros of the function $f(x) = 2 - 4 \cos x$ are the values of $x$ for which the function $f(x)$ equals zero. In other words, these values of $x$ make the function $f(x)$ equal to zero.

Q: How do we use the zeros of the function $f(x) = 2 - 4 \cos x$ to solve problems involving the function $f(x) = 2 - 4 \cos x$?

A: We use the zeros of the function $f(x) = 2 - 4 \cos x$ to solve problems involving the function $f(x) = 2 - 4 \cos x$ by substituting these values into the equation $f(x) = 2 - 4 \cos x$. This gives us the values of $x$ for which the function $f(x)$ equals zero.

Q: What are some common applications of the function $f(x) = 2 - 4 \cos x$?

A: Some common applications of the function $f(x) = 2 - 4 \cos x$ include:

• Modeling the motion of a pendulum
• Modeling the motion of a spring
• Modeling the behavior of a physical system

Q: How do we use the function $f(x) = 2 - 4 \cos x$ to model the motion of a pendulum?

A: We use the function $f(x) = 2 - 4 \cos x$ to model the motion of a pendulum by substituting the values of $x$ for which $\cos x = \frac{1}{2}$ into the equation $f(x) = 2 - 4 \cos x$. This gives us the values of $x$ for which the pendulum is in equilibrium.

Q: How do we use the function $f(x) = 2 - 4 \cos x$ to model the motion of a spring?

A: We use the function $f(x) = 2 - 4 \cos x$ to model the motion of a spring by substituting the values of $x$ for which $\cos x = \frac{1}{2}$ into the equation $f(x) = 2 - 4 \cos x$. This gives us the values of $x$ for which the spring is in equilibrium.

Q: How do we use the function $f(x) = 2 - 4 \cos x$ to model the behavior of a physical system?

A: We use the function $f(x) = 2 - 4 \cos x$ to model the behavior of a physical system by substituting the values of $x$ for which $\cos x = \frac{1}{2}$ into the equation $f(x) = 2 - 4 \cos x$. This gives us the values of $x$ for which the physical system is in equilibrium.

Q: What are some common challenges when using the function $f(x) = 2 - 4 \cos x$ to model physical systems?

A: Some common challenges when using the function $f(x) = 2 - 4 \cos x$ to model physical systems include:

• Determining the values of $x$ for which $\cos x = \frac{1}{2}$
• Substituting these values into the equation $f(x) = 2 - 4 \cos x$
• Solving the resulting equation for $x$

Q: How do we overcome these challenges when using the function $f(x) = 2 - 4 \cos x$ to model physical systems?

A: We overcome these challenges by using mathematical techniques such as substitution and solving equations. We also use physical principles such as the conservation of energy and the conservation of momentum to model the behavior of physical systems.

Q: What are some common applications of the function $f(x) = 2 - 4 \cos x$ in engineering?

A: Some common applications of the function $f(x) = 2 - 4 \cos x$ in engineering include:

• Modeling the motion of mechanical systems
• Modeling the behavior of electrical systems
• Modeling the behavior of thermal systems

Q: How do we use the function $f(x) = 2 - 4 \cos x$ to model the motion of mechanical systems?

A: We use the function $f(x) = 2 - 4 \cos x$ to model the motion of mechanical systems by substituting the values of $x$ for which $\cos x = \frac{1}{2}$ into the equation $f(x) = 2 - 4 \cos x$. This gives us the values of $x$ for which the mechanical system is in equilibrium.

Q: How do we use the function $f(x) = 2 - 4 \cos x$ to model the behavior of electrical systems?

A: We use the function $f(x) = 2 - 4 \cos x$ to model the behavior of electrical systems by substituting the values of $x$ for which $\cos x = \frac{1}{2}$ into the equation $f(x) = 2 - 4 \cos x$. This gives us the values of $x$ for which the electrical system is in equilibrium.

Q: How do we use the function $f(x) = 2 - 4 \cos x$ to model the behavior of thermal systems?

A: We use the function $f(x) = 2 - 4 \cos x$ to model the behavior of thermal systems by substituting the values of $x$ for which $\cos x = \frac{1}{2}$ into the equation $f(x) = 2 - 4 \cos x$. This gives us the values of $x$ for which the thermal system