Understanding the Components of a Cosine Function

Oct 13, 2025 by talynt.org 86 views

Understanding the Components of a Cosine Function

A cosine function is a type of trigonometric function that oscillates between a maximum and minimum value. The general form of a cosine function is given by:

y = A \cos(Bx - C) + D

where:

$A$ is the amplitude of the function, which is the distance from the midline to the maximum or minimum value.
$B$ is the frequency of the function, which is related to the period by the formula $P = \frac{2\pi}{B}$ .
$C$ is the phase shift of the function, which determines the horizontal position of the function.
$D$ is the vertical translation of the function, which determines the vertical position of the function.

Period, Amplitude, and Vertical Translation

In this problem, we are given the following information:

The period of the function is $\frac{6\pi}{5}$ .
The amplitude of the function is 5.
The vertical translation of the function is 6 units down.

Finding the Equation of the Cosine Function

To find the equation of the cosine function, we need to use the given information to determine the values of $A$ , $B$ , $C$ , and $D$ .

Period and Frequency

The period of the function is given by $P = \frac{2\pi}{B}$ . We can rearrange this formula to solve for $B$ :

B = \frac{2\pi}{P}

Substituting the given period of $\frac{6\pi}{5}$ , we get:

B = \frac{2\pi}{\frac{6\pi}{5}} = \frac{5}{3}

Amplitude

The amplitude of the function is given as 5. This means that the maximum value of the function is $A + D = 5 + 6 = 11$ , and the minimum value of the function is $-A + D = -5 + 6 = 1$ .

Vertical Translation

The vertical translation of the function is given as 6 units down. This means that the function is shifted down by 6 units, so the value of $D$ is $-6$ .

Writing the Equation of the Cosine Function

Now that we have determined the values of $A$ , $B$ , $C$ , and $D$ , we can write the equation of the cosine function:

y = 5 \cos\left(\frac{5}{3}x\right) - 6

This is the equation of the cosine function with the given period, amplitude, and vertical translation.

Conclusion

In this problem, we used the given information to determine the values of $A$ , $B$ , $C$ , and $D$ for the cosine function. We then used these values to write the equation of the cosine function. The equation of the cosine function is given by:

y = 5 \cos\left(\frac{5}{3}x\right) - 6

This equation represents a cosine function with a period of $\frac{6\pi}{5}$ , an amplitude of 5, and a vertical translation of 6 units down.

Frequently Asked Questions

Q: What is the period of the cosine function?

A: The period of the cosine function is $\frac{6\pi}{5}$ .

Q: What is the amplitude of the cosine function?

A: The amplitude of the cosine function is 5.

Q: What is the vertical translation of the cosine function?

A: The vertical translation of the cosine function is 6 units down.

Q: How do I write the equation of a cosine function with a given period, amplitude, and vertical translation?

A: To write the equation of a cosine function with a given period, amplitude, and vertical translation, you need to determine the values of $A$ , $B$ , $C$ , and $D$ using the given information. You can then use these values to write the equation of the cosine function.

Step-by-Step Solution

Step 1: Determine the value of $B$

To determine the value of $B$ , we need to use the formula $B = \frac{2\pi}{P}$ , where $P$ is the period of the function.

Step 2: Determine the value of $A$

To determine the value of $A$ , we need to use the given amplitude of the function.

Step 3: Determine the value of $D$

To determine the value of $D$ , we need to use the given vertical translation of the function.

Step 4: Write the equation of the cosine function

To write the equation of the cosine function, we need to use the values of $A$ , $B$ , $C$ , and $D$ that we determined in the previous steps.

Final Answer

The final answer is: $\boxed{y = 5 \cos\left(\frac{5}{3}x\right) - 6}$

Understanding the Components of a Cosine Function

A cosine function is a type of trigonometric function that oscillates between a maximum and minimum value. The general form of a cosine function is given by:

y = A \cos(Bx - C) + D

where:

$A$ is the amplitude of the function, which is the distance from the midline to the maximum or minimum value.
$B$ is the frequency of the function, which is related to the period by the formula $P = \frac{2\pi}{B}$ .
$C$ is the phase shift of the function, which determines the horizontal position of the function.
$D$ is the vertical translation of the function, which determines the vertical position of the function.

Q&A

Q: What is the period of the cosine function?

A: The period of the cosine function is $\frac{6\pi}{5}$ .

Q: What is the amplitude of the cosine function?

A: The amplitude of the cosine function is 5.

Q: What is the vertical translation of the cosine function?

A: The vertical translation of the cosine function is 6 units down.

Q: How do I write the equation of a cosine function with a given period, amplitude, and vertical translation?

Q: What is the difference between the period and frequency of a cosine function?

A: The period of a cosine function is the time it takes for the function to complete one full cycle, while the frequency is the number of cycles per unit time. The period and frequency are related by the formula $P = \frac{2\pi}{B}$ .

Q: How do I determine the value of $B$ for a given period?

A: To determine the value of $B$ for a given period, you can use the formula $B = \frac{2\pi}{P}$ .

Q: What is the phase shift of a cosine function?

A: The phase shift of a cosine function is the horizontal position of the function, which is determined by the value of $C$ .

Q: How do I determine the value of $C$ for a given phase shift?

A: To determine the value of $C$ for a given phase shift, you can use the formula $C = \frac{\pi}{B}$ .

Q: What is the vertical translation of a cosine function?

A: The vertical translation of a cosine function is the vertical position of the function, which is determined by the value of $D$ .

Q: How do I determine the value of $D$ for a given vertical translation?

A: To determine the value of $D$ for a given vertical translation, you can use the formula $D = -k$ , where $k$ is the vertical translation.

Q: How do I write the equation of a cosine function with a given period, amplitude, and vertical translation?

Step-by-Step Solution

Step 1: Determine the value of $B$

To determine the value of $B$ , we need to use the formula $B = \frac{2\pi}{P}$ , where $P$ is the period of the function.

Step 2: Determine the value of $A$

To determine the value of $A$ , we need to use the given amplitude of the function.

Step 3: Determine the value of $D$

To determine the value of $D$ , we need to use the given vertical translation of the function.

Step 4: Write the equation of the cosine function

To write the equation of the cosine function, we need to use the values of $A$ , $B$ , $C$ , and $D$ that we determined in the previous steps.

Final Answer

The final answer is: $\boxed{y = 5 \cos\left(\frac{5}{3}x\right) - 6}$

Conclusion

In this article, we discussed the components of a cosine function and how to write the equation of a cosine function with a given period, amplitude, and vertical translation. We also provided a step-by-step solution to determine the values of $A$ , $B$ , $C$ , and $D$ and write the equation of the cosine function.

Frequently Asked Questions

Q: What is the period of the cosine function?

A: The period of the cosine function is $\frac{6\pi}{5}$ .

Q: What is the amplitude of the cosine function?

A: The amplitude of the cosine function is 5.

Q: What is the vertical translation of the cosine function?

A: The vertical translation of the cosine function is 6 units down.