On The Axes Below, Make An Appropriate Scale And Graph Exactly One Cycle Of The Trigonometric Function $y=12 \sin \frac{5}{2} X$.

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Introduction

Trigonometric functions are a fundamental concept in mathematics, and graphing them is an essential skill for any student of mathematics. In this article, we will focus on graphing the trigonometric function $y=12 \sin \frac{5}{2} x$. We will start by understanding the properties of the sine function, then create an appropriate scale and graph exactly one cycle of the given function.

Understanding the Sine Function

The sine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. The sine function has a period of 2π2\pi, which means that it repeats itself every 2π2\pi units.

Graphing the Sine Function

To graph the sine function, we need to create a coordinate system with the x-axis representing the angle and the y-axis representing the sine value. We can start by plotting the point (0, 0) and then use the properties of the sine function to plot the rest of the graph.

Graphing the Trigonometric Function $y=12 \sin \frac{5}{2} x$

To graph the trigonometric function $y=12 \sin \frac{5}{2} x$, we need to create an appropriate scale and graph exactly one cycle of the function. The function has a period of 2π52=4π5\frac{2\pi}{\frac{5}{2}} = \frac{4\pi}{5}, which means that it repeats itself every 4π5\frac{4\pi}{5} units.

Creating an Appropriate Scale

To create an appropriate scale, we need to determine the amplitude and period of the function. The amplitude of the function is 12, which means that the graph will oscillate between -12 and 12. The period of the function is 4π5\frac{4\pi}{5}, which means that the graph will repeat itself every 4π5\frac{4\pi}{5} units.

Graphing Exactly One Cycle

To graph exactly one cycle of the function, we need to plot the graph over an interval of 4π5\frac{4\pi}{5} units. We can start by plotting the point (0, 0) and then use the properties of the sine function to plot the rest of the graph.

Key Features of the Graph

The graph of the trigonometric function $y=12 \sin \frac{5}{2} x$ has several key features. The graph oscillates between -12 and 12, with a period of 4π5\frac{4\pi}{5} units. The graph has a maximum value of 12 and a minimum value of -12.

Conclusion

Graphing trigonometric functions is an essential skill for any student of mathematics. In this article, we have focused on graphing the trigonometric function $y=12 \sin \frac{5}{2} x$. We have created an appropriate scale and graphed exactly one cycle of the function. The graph has several key features, including an amplitude of 12 and a period of 4π5\frac{4\pi}{5} units.

Key Takeaways

  • The sine function is a periodic function that oscillates between -1 and 1.
  • The sine function has a period of 2π2\pi, which means that it repeats itself every 2π2\pi units.
  • The trigonometric function $y=12 \sin \frac{5}{2} x$ has a period of 4π5\frac{4\pi}{5} units.
  • The graph of the trigonometric function $y=12 \sin \frac{5}{2} x$ oscillates between -12 and 12.

Further Reading

For further reading on graphing trigonometric functions, we recommend the following resources:

  • "Graphing Trigonometric Functions" by Math Open Reference
  • "Trigonometric Functions" by Khan Academy
  • "Graphing Trigonometric Functions" by Purplemath

References

  • "Trigonometry" by Michael Corral
  • "Graphing Trigonometric Functions" by James Stewart
  • "Trigonometric Functions" by Wolfram MathWorld

Introduction

Graphing trigonometric functions is a fundamental concept in mathematics, and it can be a challenging task for many students. In this article, we will answer some of the most frequently asked questions about graphing trigonometric functions.

Q: What is the period of a trigonometric function?

A: The period of a trigonometric function is the length of one complete cycle of the function. For example, the period of the sine function is 2π2\pi, which means that it repeats itself every 2π2\pi units.

Q: How do I determine the amplitude of a trigonometric function?

A: The amplitude of a trigonometric function is the maximum value that the function reaches. For example, the amplitude of the function $y=12 \sin \frac{5}{2} x$ is 12, which means that the graph will oscillate between -12 and 12.

Q: How do I graph a trigonometric function?

A: To graph a trigonometric function, you need to create a coordinate system with the x-axis representing the angle and the y-axis representing the function value. You can start by plotting the point (0, 0) and then use the properties of the function to plot the rest of the graph.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions, but they have different properties. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle, while the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Q: How do I graph a trigonometric function with a period other than 2π2\pi?

A: To graph a trigonometric function with a period other than 2π2\pi, you need to use the formula for the period of the function. For example, if the function has a period of 4π5\frac{4\pi}{5}, you can graph it over an interval of 4π5\frac{4\pi}{5} units.

Q: What is the relationship between the sine and cosine functions?

A: The sine and cosine functions are related by the Pythagorean identity, which states that sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1. This means that the sine and cosine functions are complementary, and they can be used to find each other's values.

Q: How do I graph a trigonometric function with a vertical shift?

A: To graph a trigonometric function with a vertical shift, you need to add or subtract a constant value from the function. For example, if the function is $y=12 \sin \frac{5}{2} x + 3$, you can graph it by adding 3 to the graph of the function $y=12 \sin \frac{5}{2} x$.

Q: What is the difference between the tangent and cotangent functions?

A: The tangent and cotangent functions are both trigonometric functions, but they have different properties. The tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle, while the cotangent function is defined as the ratio of the length of the adjacent side to the length of the opposite side.

Conclusion

Graphing trigonometric functions is a fundamental concept in mathematics, and it can be a challenging task for many students. In this article, we have answered some of the most frequently asked questions about graphing trigonometric functions. We hope that this article has been helpful in clarifying some of the concepts and techniques involved in graphing trigonometric functions.

Key Takeaways

  • The period of a trigonometric function is the length of one complete cycle of the function.
  • The amplitude of a trigonometric function is the maximum value that the function reaches.
  • The sine and cosine functions are both trigonometric functions, but they have different properties.
  • The tangent and cotangent functions are both trigonometric functions, but they have different properties.

Further Reading

For further reading on graphing trigonometric functions, we recommend the following resources:

  • "Graphing Trigonometric Functions" by Math Open Reference
  • "Trigonometric Functions" by Khan Academy
  • "Graphing Trigonometric Functions" by Purplemath

References

  • "Trigonometry" by Michael Corral
  • "Graphing Trigonometric Functions" by James Stewart
  • "Trigonometric Functions" by Wolfram MathWorld