Answer the following questions by including the steps and formulas.

(a)

Write the equation sine or cosine function with the given properties:

Midline=5, amplitude = 13, period = 2π, phase shift =0


To write the equation of a sine or cosine function with the given properties (midline = 5, amplitude = 13, period = \(2\pi\), and phase shift = 0), we need to understand the general forms of these functions.

The general form of the sine function is:
\[ y = A \sin(B(x - C)) + D \]

The general form of the cosine function is:
\[ y = A \cos(B(x - C)) + D \]

Where:
- \(A\) is the amplitude
- \(B\) affects the period of the function
- \(C\) is the phase shift
- \(D\) is the midline (vertical shift)

Given the properties:
- Midline (\(D\)) = 5
- Amplitude (\(A\)) = 13
- Period = \(2\pi\)
- Phase shift (\(C\)) = 0

**Step-by-Step Solution:**

1. **Determine the amplitude \(A\):**
   \[ A = 13 \]

2. **Determine the midline \(D\):**
   \[ D = 5 \]

3. **Determine the value of \(B\) using the period:**
   The period of the sine or cosine function is given by:
   \[ \text{Period} = \frac{2\pi}{B} \]
   Given the period is \(2\pi\):
   \[ 2\pi = \frac{2\pi}{B} \]
   Solving for \(B\):
   \[ B = 1 \]

4. **Determine the phase shift \(C\):**
   Given the phase shift is 0:
   \[ C = 0 \]

Now, substituting these values into the general form:

### Sine Function:
\[ y = 13 \sin(1 \cdot (x - 0)) + 5 \]
Simplifying:
\[ y = 13 \sin(x) + 5 \]

### Cosine Function:
\[ y = 13 \cos(1 \cdot (x - 0)) + 5 \]
Simplifying:
\[ y = 13 \cos(x) + 5 \]

Thus, the equations are:
- For the sine function: \[ y = 13 \sin(x) + 5 \]
- For the cosine function: \[ y = 13 \cos(x) + 5 \]

Do you want details or have any questions?

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**Related Questions:**
1. How do you determine the period of a sine or cosine function?
2. What effect does the amplitude have on the graph of a sine or cosine function?
3. How can you determine the phase shift of a trigonometric function from its equation?
4. What is the midline of a trigonometric function and how do you find it?
5. How would you modify the function if the period was \(4\pi\) instead of \(2\pi\)?
6. Can you explain how to shift a sine or cosine function vertically and horizontally?
7. What is the relationship between the frequency and period of a trigonometric function?
8. How do you graph a sine function with the given amplitude, period, and phase shift?

**Tip:** To find the phase shift, look for the horizontal shift in the function's argument, \(x - C\). A positive \(C\) shifts the graph to the right, and a negative \(C\) shifts it to the left.

Learn how to write the equation of sine and cosine functions with specific properties: midline = 5, amplitude = 13, period = 2π, and phase shift = 0. Understand the general forms and steps involved in determining these trigonometric functions. Suitable for advanced high school students and college-level learners.

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