Write The Equation Of A Sine Function That Has The Following Characteristics:- Amplitude: 7- Period: $5 \pi$- Phase Shift: $\frac{1}{8}$

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Introduction

In mathematics, a sine function is a fundamental concept in trigonometry, used to describe periodic phenomena. The general equation of a sine function is given by:

y=Asin⁑(Bxβˆ’C)+Dy = A \sin(Bx - C) + D

where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.

In this article, we will discuss how to write the equation of a sine function with the following characteristics:

  • Amplitude: 7
  • Period: $5 \pi$
  • Phase shift: $\frac{1}{8}$

Understanding the Characteristics

Amplitude

The amplitude of a sine function is the maximum value it can attain. In this case, the amplitude is given as 7. This means that the function will oscillate between -7 and 7.

Period

The period of a sine function is the time taken by the function to complete one full cycle. In this case, the period is given as $5 \pi$. This means that the function will complete one full cycle in $5 \pi$ units of time.

Phase Shift

The phase shift of a sine function is the horizontal shift of the function from its standard position. In this case, the phase shift is given as $\frac{1}{8}$. This means that the function will be shifted $\frac{1}{8}$ units to the right.

Writing the Equation

To write the equation of a sine function with the given characteristics, we need to use the general equation of a sine function:

y=Asin⁑(Bxβˆ’C)+Dy = A \sin(Bx - C) + D

We are given that the amplitude is 7, so we can substitute A = 7 into the equation:

y=7sin⁑(Bxβˆ’C)+Dy = 7 \sin(Bx - C) + D

We are also given that the period is $5 \pi$, so we can use the formula:

B=2Ο€periodB = \frac{2\pi}{\text{period}}

to find the value of B:

B=2Ο€5Ο€=25B = \frac{2\pi}{5\pi} = \frac{2}{5}

Substituting this value of B into the equation, we get:

y=7sin⁑(25xβˆ’C)+Dy = 7 \sin\left(\frac{2}{5}x - C\right) + D

We are also given that the phase shift is $\frac{1}{8}$, so we can substitute C = $\frac{1}{8}$ into the equation:

y=7sin⁑(25xβˆ’18)+Dy = 7 \sin\left(\frac{2}{5}x - \frac{1}{8}\right) + D

To find the value of D, we need to use the fact that the function passes through the point (0, 0). This means that when x = 0, y = 0. Substituting these values into the equation, we get:

0=7sin⁑(βˆ’18)+D0 = 7 \sin\left(-\frac{1}{8}\right) + D

Solving for D, we get:

D=βˆ’7sin⁑(βˆ’18)D = -7 \sin\left(-\frac{1}{8}\right)

Substituting this value of D into the equation, we get:

y=7sin⁑(25xβˆ’18)βˆ’7sin⁑(βˆ’18)y = 7 \sin\left(\frac{2}{5}x - \frac{1}{8}\right) - 7 \sin\left(-\frac{1}{8}\right)

Simplifying the Equation

To simplify the equation, we can use the fact that $\sin(-x) = -\sin(x)$. Substituting this into the equation, we get:

y=7sin⁑(25xβˆ’18)+7sin⁑(18)y = 7 \sin\left(\frac{2}{5}x - \frac{1}{8}\right) + 7 \sin\left(\frac{1}{8}\right)

This is the final equation of the sine function with the given characteristics.

Conclusion

In this article, we discussed how to write the equation of a sine function with the following characteristics:

  • Amplitude: 7
  • Period: $5 \pi$
  • Phase shift: $\frac{1}{8}$

We used the general equation of a sine function and substituted the given values to find the final equation. The equation is:

y=7sin⁑(25xβˆ’18)+7sin⁑(18)y = 7 \sin\left(\frac{2}{5}x - \frac{1}{8}\right) + 7 \sin\left(\frac{1}{8}\right)

This equation can be used to model periodic phenomena with the given characteristics.

Example Use Cases

The equation of a sine function with the given characteristics can be used to model a variety of real-world phenomena, such as:

  • The motion of a pendulum with a period of $5 \pi$ and an amplitude of 7.
  • The vibration of a spring with a frequency of $\frac{2}{5}$ and a phase shift of $\frac{1}{8}$.
  • The oscillation of a mass on a spring with a period of $5 \pi$ and an amplitude of 7.

These are just a few examples of the many possible use cases for the equation of a sine function with the given characteristics.

Glossary of Terms

  • Amplitude: The maximum value of a sine function.
  • Period: The time taken by a sine function to complete one full cycle.
  • Phase shift: The horizontal shift of a sine function from its standard position.
  • Frequency: The number of cycles of a sine function per unit time.
  • Vertical shift: The vertical shift of a sine function from its standard position.

These terms are used throughout the article to describe the characteristics of the sine function.

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Q: What is the general equation of a sine function?

A: The general equation of a sine function is:

y=Asin⁑(Bxβˆ’C)+Dy = A \sin(Bx - C) + D

where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.

Q: How do I find the value of B in the equation of a sine function?

A: To find the value of B, you can use the formula:

B=2Ο€periodB = \frac{2\pi}{\text{period}}

where period is the time taken by the function to complete one full cycle.

Q: What is the difference between frequency and period?

A: Frequency and period are related but distinct concepts. Frequency is the number of cycles of a sine function per unit time, while period is the time taken by the function to complete one full cycle. The relationship between frequency and period is given by:

frequency=1period\text{frequency} = \frac{1}{\text{period}}

Q: How do I find the value of C in the equation of a sine function?

A: To find the value of C, you can use the formula:

C=phaseΒ shiftfrequencyC = \frac{\text{phase shift}}{\text{frequency}}

where phase shift is the horizontal shift of the function from its standard position.

Q: What is the vertical shift in the equation of a sine function?

A: The vertical shift in the equation of a sine function is the value of D, which is the value of the function at x = 0.

Q: How do I find the value of D in the equation of a sine function?

A: To find the value of D, you can use the fact that the function passes through the point (0, 0). This means that when x = 0, y = 0. Substituting these values into the equation, you can solve for D.

Q: Can I use the equation of a sine function to model real-world phenomena?

A: Yes, the equation of a sine function can be used to model a variety of real-world phenomena, such as the motion of a pendulum, the vibration of a spring, and the oscillation of a mass on a spring.

Q: What are some common applications of the equation of a sine function?

A: Some common applications of the equation of a sine function include:

  • Modeling the motion of a pendulum
  • Modeling the vibration of a spring
  • Modeling the oscillation of a mass on a spring
  • Modeling the tides
  • Modeling the sound waves

Q: How do I simplify the equation of a sine function?

A: To simplify the equation of a sine function, you can use various mathematical techniques, such as:

  • Using the identity $\sin(-x) = -\sin(x)$
  • Using the identity $\cos(x) = \sin\left(\frac{\pi}{2} - x\right)$
  • Using the identity $\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)$

Q: Can I use the equation of a sine function to model periodic phenomena with different characteristics?

A: Yes, the equation of a sine function can be used to model periodic phenomena with different characteristics, such as:

  • Different amplitudes
  • Different periods
  • Different phase shifts
  • Different frequencies

By adjusting the values of A, B, C, and D, you can model a wide range of periodic phenomena.

Q: How do I graph the equation of a sine function?

A: To graph the equation of a sine function, you can use various graphing techniques, such as:

  • Plotting the function on a coordinate plane
  • Using a graphing calculator or software
  • Using a computer program to generate a graph

By graphing the equation of a sine function, you can visualize the behavior of the function and gain a deeper understanding of its characteristics.