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what is the equation for the following sine function

To find the equation of the sine function represented by the graph, we need to determine the amplitude, period, phase shift, and vertical shift of the function.

1. **Amplitude**: The amplitude is the distance from the midline to the peak. Here, the peak is at \( y = 2 \) and the trough is at \( y = -2 \). The amplitude \( A \) is:
   \[
   A = 2
   \]

2. **Period**: The period \( T \) is the distance along the x-axis for one complete cycle. Observing the graph, the function completes one cycle from \( x = 0 \) to \( x = 180 \). Hence, the period is:
   \[
   T = 180
   \]

   The standard period of \( \sin(x) \) is \( 2\pi \), so the period adjustment factor \( B \) is:
   \[
   B = \frac{2\pi}{T} = \frac{2\pi}{180} = \frac{\pi}{90}
   \]

3. **Phase Shift**: There appears to be no horizontal shift as the sine function starts at \( x = 0 \) and the midline at \( y = 0 \).

4. **Vertical Shift**: The midline of the sine wave is at \( y = 0 \), indicating there is no vertical shift \( D \).

Therefore, the equation of the sine function is:
\[
y = 2 \sin\left(\frac{\pi}{90} x\right)
\]

Would you like any further details or have any questions?

Here are 5 more questions you can ask:
1. How do you find the phase shift of a trigonometric function?
2. What is the difference between sine and cosine functions?
3. How do you graph a cosine function?
4. How can you determine the frequency of a trigonometric function from its equation?
5. How do you convert between degrees and radians?

**Tip:** When finding the equation of a trigonometric function from a graph, always start by identifying the amplitude and period, as they are fundamental to the function's form.

Discover how to find the equation of a sine function with an amplitude of 2 and a period of 180 degrees. This problem involves trigonometric concepts such as amplitude, period, and sine function equations. Suitable for high school students studying trigonometry.

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