Find an equation for a sinusoidal function that has a period 2(pi), amplitude 1,and contains the point (pi/2,-3)

Let's find the equation of the sinusoidal function with the given properties.

### Step 1: General Form of the Sinusoidal Function

The general form of a sinusoidal function is:

\[
y = A \sin(Bx + C) + D
\]

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

### Step 2: Identify the Given Parameters

1. **Amplitude** (\( A \)): Given as 1, so \( A = 1 \).
2. **Period**: Given as \( 2\pi \). The period is given by \( \frac{2\pi}{|B|} \). Since the period is \( 2\pi \), we have:
   \[
   \frac{2\pi}{|B|} = 2\pi
   \]
   Simplifying gives \( |B| = 1 \). So, \( B = 1 \).
3. **Point**: The function passes through the point \( \left(\frac{\pi}{2}, -3\right) \).

### Step 3: Set Up the Equation

Now we substitute the known values into the equation:

\[
y = \sin(x + C) + D
\]

### Step 4: Use the Point \( \left(\frac{\pi}{2}, -3\right) \) to Find \( C \) and \( D \)

Substitute \( x = \frac{\pi}{2} \) and \( y = -3 \) into the equation:

\[
-3 = \sin\left(\frac{\pi}{2} + C\right) + D
\]

This simplifies to:

\[
-3 = 1 + D \quad \text{(since \( \sin\left(\frac{\pi}{2} + C\right) \) reaches its maximum value of 1 when \( C = 0 \))}
\]

\[
D = -4
\]

Now the equation is:

\[
y = \sin(x + C) - 4
\]

### Step 5: Determine \( C \) Using the Known Point

Given that \( \sin\left(\frac{\pi}{2} + C\right) = 1 \), and the sine function equals 1 at \( \frac{\pi}{2} \), \( C \) should be 0. 

Therefore, the equation becomes:

\[
y = \sin(x) - 4
\]

### Final Answer:

The equation of the sinusoidal function that meets all the given conditions is:

\[
\boxed{y = \sin(x) - 4}
\]

Would you like more details or have any questions?

### Relative Questions:
1. How does the vertical shift \( D \) affect the graph of a sinusoidal function?
2. What would the equation look like if the function had a different phase shift \( C \)?
3. How can you determine the maximum and minimum values of a sinusoidal function?
4. What happens to the function \( y = \sin(x) \) if the amplitude is changed to 2?
5. How does the sine function compare to the cosine function in terms of phase shifts?

### Tip:
Remember that the vertical shift \( D \) moves the entire graph up or down, depending on its sign. It represents the midline of the sinusoidal wave.

Learn how to find the equation of a sinusoidal function with a period of 2π, amplitude of 1, passing through the point (π/2, -3). This problem involves trigonometric concepts and their applications. Suitable for students in Grades 10-12.

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