Determine The { X $}$-intercepts Of The Function { G(x) = 2 \sin (2x) + 1 $}$ On The Interval { [0, 2\pi]$}$.A. { \left{\frac{7\pi}{6}, \frac{11\pi}{6}\right}$} B . \[ B. \[ B . \[ \left{\frac{7\pi}{12},

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Introduction

In mathematics, the x-intercepts of a function are the points where the graph of the function crosses the x-axis. These points are also known as the roots or zeros of the function. In this article, we will determine the x-intercepts of the function g(x) = 2 sin(2x) + 1 on the interval [0, 2Ï€].

Understanding the Function

The given function is g(x) = 2 sin(2x) + 1. This is a trigonometric function that involves the sine of 2x multiplied by 2 and then added to 1. To find the x-intercepts, we need to set the function equal to zero and solve for x.

Setting the Function Equal to Zero

To find the x-intercepts, we set the function g(x) equal to zero:

2 sin(2x) + 1 = 0

Solving for x

Now, we need to solve for x. We can start by isolating the sine term:

2 sin(2x) = -1

Divide both sides by 2:

sin(2x) = -1/2

Using the Unit Circle

To solve the equation sin(2x) = -1/2, we can use the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine function is defined as the y-coordinate of a point on the unit circle.

Finding the Angles

We need to find the angles on the unit circle that correspond to a sine value of -1/2. These angles are:

2x = 7Ï€/6 and 2x = 11Ï€/6

Solving for x

Now, we can solve for x by dividing both sides by 2:

x = 7Ï€/12 and x = 11Ï€/12

Checking the Solutions

We need to check if these solutions are within the given interval [0, 2Ï€]. Both solutions are within the interval, so they are valid x-intercepts.

Conclusion

In conclusion, the x-intercepts of the function g(x) = 2 sin(2x) + 1 on the interval [0, 2Ï€] are:

x = 7Ï€/12 and x = 11Ï€/12

These solutions are valid because they are within the given interval and satisfy the equation g(x) = 0.

Answer

The correct answer is:

A. {7Ï€/6, 11Ï€/6}

However, the solutions we found are x = 7Ï€/12 and x = 11Ï€/12, which are not among the options. This suggests that the question may be incorrect or that there is a mistake in the solution.

Discussion

This problem involves finding the x-intercepts of a trigonometric function. The function g(x) = 2 sin(2x) + 1 is a periodic function with a period of π. The x-intercepts occur when the function crosses the x-axis, which happens when the sine term is equal to -1/2.

The unit circle is a useful tool for solving trigonometric equations. By using the unit circle, we can find the angles that correspond to a given sine value. In this case, we found the angles 2x = 7Ï€/6 and 2x = 11Ï€/6, which correspond to a sine value of -1/2.

The solutions x = 7Ï€/12 and x = 11Ï€/12 are valid because they are within the given interval and satisfy the equation g(x) = 0. However, these solutions are not among the options, which suggests that the question may be incorrect or that there is a mistake in the solution.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by James Stewart

Keywords

  • Trigonometric function
  • x-intercepts
  • Unit circle
  • Sine function
  • Periodic function
  • Calculus
  • Mathematics
    Q&A: Determining the x-Intercepts of a Trigonometric Function ===========================================================

Q: What is the x-intercept of a function?

A: The x-intercept of a function is the point where the graph of the function crosses the x-axis. This point is also known as the root or zero of the function.

Q: How do you find the x-intercepts of a trigonometric function?

A: To find the x-intercepts of a trigonometric function, you need to set the function equal to zero and solve for x. This involves using trigonometric identities and equations to isolate the variable x.

Q: What is the unit circle and how is it used in trigonometry?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is used to define the trigonometric functions sine, cosine, and tangent. By using the unit circle, you can find the angles that correspond to a given sine value.

Q: How do you use the unit circle to solve trigonometric equations?

A: To use the unit circle to solve trigonometric equations, you need to identify the angle that corresponds to the given sine value. This involves using the unit circle to find the angle that has a sine value equal to the given value.

Q: What is the period of a trigonometric function?

A: The period of a trigonometric function is the distance between two consecutive points on the graph of the function that have the same y-coordinate. For example, the period of the sine function is 2Ï€.

Q: How do you find the x-intercepts of a periodic function?

A: To find the x-intercepts of a periodic function, you need to find the points on the graph of the function that have a y-coordinate of zero. This involves using the period of the function to identify the points on the graph that have the same y-coordinate.

Q: What is the difference between the x-intercepts of a trigonometric function and the x-intercepts of a polynomial function?

A: The x-intercepts of a trigonometric function are the points on the graph of the function that have a y-coordinate of zero. The x-intercepts of a polynomial function are the points on the graph of the function that have a y-coordinate of zero. However, the x-intercepts of a trigonometric function are typically found using trigonometric identities and equations, while the x-intercepts of a polynomial function are typically found using algebraic methods.

Q: Can you give an example of a trigonometric function and its x-intercepts?

A: Yes, the function g(x) = 2 sin(2x) + 1 is a trigonometric function. Its x-intercepts are x = 7Ï€/12 and x = 11Ï€/12.

Q: How do you check if a solution is valid?

A: To check if a solution is valid, you need to verify that it satisfies the original equation and that it is within the given interval.

Q: What is the importance of finding the x-intercepts of a function?

A: Finding the x-intercepts of a function is important because it helps you understand the behavior of the function and its graph. It also helps you identify the points on the graph where the function crosses the x-axis.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by James Stewart

Keywords

  • Trigonometric function
  • x-intercepts
  • Unit circle
  • Sine function
  • Periodic function
  • Calculus
  • Mathematics