What Are The Zeros Of The Function F ( X ) = 2 − 4 Sin ⁡ X F(x)=2-4 \sin X F ( X ) = 2 − 4 Sin 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=\frac{\pi}{3}+2 \pi

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Understanding the Problem

The problem asks us to find the zeros of the function $f(x)=2-4 \sin x$. In other words, we need to find the values of $x$ for which the function $f(x)$ equals zero. This is a classic problem in mathematics, and it requires a good understanding of trigonometric functions and their properties.

Recalling the Definition of Sine Function

Before we proceed, let's recall the definition of the sine function. The sine function is defined as the ratio of the length of the side opposite a given angle to the length of the hypotenuse in a right-angled triangle. In other words, if we have a right-angled triangle with an angle $\theta$, then the sine of $\theta$ is equal to the length of the side opposite $\theta$ divided by the length of the hypotenuse.

Graph of the Sine Function

The graph of the sine function is a periodic curve that oscillates between $-1$ and $1$. The graph has a period of $2\pi$, which means that it repeats itself every $2\pi$ units. The graph also has a maximum value of $1$ and a minimum value of $-1$.

Finding the Zeros of the Function

To find the zeros of the function $f(x)=2-4 \sin x$, we need to set the function equal to zero and solve for $x$. In other words, we need to find the values of $x$ for which $2-4 \sin x=0$. This can be rewritten as $-4 \sin x=-2$, or $\sin x=\frac{1}{2}$.

Solving the Equation

To solve the equation $\sin x=\frac{1}{2}$, we need to find the values of $x$ for which the sine function equals $\frac{1}{2}$. This can be done by using the unit circle or by using a calculator. The values of $x$ for which the sine function equals $\frac{1}{2}$ are $x=\frac{\pi}{6}$ and $x=\frac{5\pi}{6}$.

General Solution

However, we are not done yet. We need to find the general solution of the equation $\sin x=\frac{1}{2}$. This can be done by adding multiples of $2\pi$ to the values of $x$ that we found earlier. In other words, the general solution of the equation $\sin x=\frac{1}{2}$ is $x=\frac{\pi}{6}+2\pi n$ and $x=\frac{5\pi}{6}+2\pi n$, where $n$ is an integer.

Conclusion

In conclusion, the zeros of the function $f(x)=2-4 \sin x$ are $x=\frac{\pi}{6}+2\pi n$ and $x=\frac{5\pi}{6}+2\pi n$, where $n$ is an integer. This is a classic problem in mathematics, and it requires a good understanding of trigonometric functions and their properties.

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 find the zeros of a trigonometric function. It requires a good understanding of the properties of the sine function and how to use them to solve equations. The problem is also a good example of how to use the unit circle to find the values of $x$ for which the sine function equals a given value.

Related Problems

This problem is related to other problems in mathematics that involve finding the zeros of trigonometric functions. Some examples of related problems include:

• Finding the zeros of the function $f(x)=\cos x$
• Finding the zeros of the function $f(x)=\tan x$
• Finding the zeros of the function $f(x)=\csc x$

Applications

This problem has many applications in mathematics and science. Some examples of applications include:

• Finding the zeros of a trigonometric function is an important step in solving many problems in mathematics and science.
• The zeros of a trigonometric function can be used to find the values of $x$ for which a given function equals a given value.
• The zeros of a trigonometric function can be used to find the values of $x$ for which a given function has a maximum or minimum value.

Conclusion

In conclusion, the zeros of the function $f(x)=2-4 \sin x$ are $x=\frac{\pi}{6}+2\pi n$ and $x=\frac{5\pi}{6}+2\pi n$, where $n$ is an integer. This is a classic problem in mathematics, and it requires a good understanding of trigonometric functions and their properties.

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

A: The function $f(x)=2-4 \sin x$ is a trigonometric function that involves the sine function. The function has a period of $2\pi$ and oscillates between $-2$ and $2$.

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

A: The zeros of the function $f(x)=2-4 \sin x$ are the values of $x$ for which the function equals zero. In other words, we need to find the values of $x$ for which $2-4 \sin x=0$. This can be rewritten as $-4 \sin x=-2$, or $\sin x=\frac{1}{2}$.

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

A: To find the zeros of the function $f(x)=2-4 \sin x$, we need to set the function equal to zero and solve for $x$. In other words, we need to find the values of $x$ for which $2-4 \sin x=0$. This can be done by using the unit circle or by using a calculator.

Q: What are the values of $x$ for which the sine function equals $\frac{1}{2}$?

A: The values of $x$ for which the sine function equals $\frac{1}{2}$ are $x=\frac{\pi}{6}$ and $x=\frac{5\pi}{6}$.

Q: How do we find the general solution of the equation $\sin x=\frac{1}{2}$?

A: To find the general solution of the equation $\sin x=\frac{1}{2}$, we need to add multiples of $2\pi$ to the values of $x$ that we found earlier. In other words, the general solution of the equation $\sin x=\frac{1}{2}$ is $x=\frac{\pi}{6}+2\pi n$ and $x=\frac{5\pi}{6}+2\pi n$, where $n$ is an integer.

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

A: The zeros of the function $f(x)=2-4 \sin x$ are $x=\frac{\pi}{6}+2\pi n$ and $x=\frac{5\pi}{6}+2\pi n$, where $n$ is an integer.

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

A: The period of the function $f(x)=2-4 \sin x$ is $2\pi$.

Q: What are the maximum and minimum values of the function $f(x)=2-4 \sin x$?

A: The maximum value of the function $f(x)=2-4 \sin x$ is $2$ and the minimum value is $-2$.

Q: How do we use the unit circle to find the values of $x$ for which the sine function equals a given value?

A: To use the unit circle to find the values of $x$ for which the sine function equals a given value, we need to draw a unit circle and mark the points on the circle that correspond to the given value of the sine function. We can then use the coordinates of these points to find the values of $x$ for which the sine function equals the given value.

Q: What are some applications of finding the zeros of a trigonometric function?

A: Some applications of finding the zeros of a trigonometric function include:

• Finding the values of $x$ for which a given function equals a given value
• Finding the values of $x$ for which a given function has a maximum or minimum value
• Solving equations involving trigonometric functions
• Finding the zeros of a trigonometric function is an important step in solving many problems in mathematics and science.

Q: What are some related problems to finding the zeros of the function $f(x)=2-4 \sin x$?

A: Some related problems to finding the zeros of the function $f(x)=2-4 \sin x$ include:

• Finding the zeros of the function $f(x)=\cos x$
• Finding the zeros of the function $f(x)=\tan x$
• Finding the zeros of the function $f(x)=\csc x$

Q: What are some real-world applications of finding the zeros of a trigonometric function?

A: Some real-world applications of finding the zeros of a trigonometric function include:

• Finding the values of $x$ for which a given function equals a given value in physics and engineering
• Finding the values of $x$ for which a given function has a maximum or minimum value in economics and finance
• Solving equations involving trigonometric functions in computer science and programming.