y = cos x solve using (275, 175) (400,275) (500,175)

To solve for the function \( y = \cos(x) \) using the points \((275, 175)\), \((400, 275)\), and \((500, 175)\), we'll follow a systematic approach to determine the transformation of the cosine function that fits these points.

### Step 1: Understand the Cosine Function
The general form of the cosine function is:
\[
y = a \cos(bx + c) + d
\]
where:
- \(a\) is the amplitude.
- \(b\) affects the period.
- \(c\) is the phase shift.
- \(d\) is the vertical shift.

### Step 2: Analyze the Given Points
- **Point 1**: \((275, 175)\)
- **Point 2**: \((400, 275)\)
- **Point 3**: \((500, 175)\)

### Step 3: Identify Key Features from the Points
- The \(y\) value at \((400, 275)\) is the maximum, which suggests that this point represents the peak of the cosine wave.
- The \(y\) values at \((275, 175)\) and \((500, 175)\) are identical, suggesting that these points might correspond to positions where the cosine function has the same value, indicating symmetry.

### Step 4: Determine the Amplitude and Vertical Shift
1. **Amplitude \(a\)**: 
   - The amplitude is half the difference between the maximum and minimum \(y\) values.
   \[
   a = \frac{275 - 175}{2} = 50
   \]

2. **Vertical Shift \(d\)**: 
   - The vertical shift is the average of the maximum and minimum \(y\) values.
   \[
   d = \frac{275 + 175}{2} = 225
   \]

So far, our function is:
\[
y = 50 \cos(bx + c) + 225
\]

### Step 5: Determine the Period and Phase Shift
- The points \((275, 175)\) and \((500, 175)\) correspond to \(y\) values equal to the mean, suggesting these points might be a quarter and three-quarters through the period of the cosine function.
- The full period \(T\) occurs between the first and third points, covering a \(2\pi\) range. 

Given:
- \(T = 500 - 275 = 225\)

The period formula is:
\[
\text{Period} = \frac{2\pi}{b}
\]

So:
\[
b = \frac{2\pi}{225}
\]

### Step 6: Determine the Phase Shift \(c\)
- The maximum point occurs at \(x = 400\). For a standard cosine function, this corresponds to \(x = 0\), so the phase shift should make \(bx + c = 0\) at \(x = 400\).

So:
\[
400 \cdot \frac{2\pi}{225} + c = 0
\]
\[
c = - \frac{800\pi}{225}
\]

### Final Equation
Putting everything together, the cosine function that fits the given points is approximately:
\[
y = 50 \cos\left(\frac{2\pi}{225} x - \frac{800\pi}{225}\right) + 225
\]

### Summary:
The final function is:
\[
y = 50 \cos\left(\frac{2\pi}{225} (x - 400)\right) + 225
\]
This function describes the cosine wave that fits the given points.

Do you have any questions or would you like to see more details?

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**Related Questions:**
1. How do you derive the amplitude of a trigonometric function from points?
2. What steps are involved in determining the period of a cosine function from given points?
3. How do you find the vertical shift in a trigonometric function?
4. How does phase shift affect the graph of a cosine function?
5. How can the cosine function be used to model real-world periodic phenomena?
6. What is the significance of the period \(T\) in a trigonometric function?
7. How does changing the amplitude affect the appearance of the cosine graph?
8. What role does the vertical shift \(d\) play in adjusting the baseline of the graph?

**Tip:** When adjusting trigonometric functions to fit data points, carefully observe the symmetry and key points, like maximums and minimums, to deduce transformations.

Learn how to solve the cosine function y = cos(x) using the points (275, 175), (400, 275), and (500, 175). This detailed explanation covers determining amplitude, period, phase shift, and vertical shift to fit the given data points. Suitable for advanced high school students and anyone interested in trigonometry.

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