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

To find the equation of a sinusoidal function with the given properties, we start by considering the general form of a sinusoidal function:

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

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

Given the problem's conditions:
- The amplitude \( A = 1 \).
- The period \( T = 2\pi \), which relates to \( B \) by the equation \( T = \frac{2\pi}{B} \). Therefore, \( B = 1 \).
- The vertical shift \( D = 0 \) because there's no mention of a shift up or down.

Thus, the equation simplifies to:

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

Next, we use the given point \((- \frac{\pi}{2}, 1)\) to find the phase shift \( C \):

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

The sine function equals 1 at \( \frac{\pi}{2} \), so:

\[
-\frac{\pi}{2} + C = \frac{\pi}{2}
\]

Solving for \( C \):

\[
C = \frac{\pi}{2} + \frac{\pi}{2} = \pi
\]

Therefore, the equation of the sinusoidal function is:

\[
y = \sin(x + \pi)
\]

Alternatively, using the identity \( \sin(x + \pi) = -\sin(x) \), the function can also be written as:

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

Would you like to dive deeper into this, or do you have any other questions?

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Here are some related questions you might find helpful:
1. How do you find the phase shift in a sinusoidal function?
2. What effect does the amplitude have on the graph of a sinusoidal function?
3. How do you determine the vertical shift in a sinusoidal function?
4. Can the period of a sinusoidal function ever be less than \( 2\pi \)?
5. How do different values of \( B \) affect the period of a sinusoidal function?

**Tip:** Always double-check the given point in the context of the calculated function to ensure accuracy!

Discover how to find the equation of a sinusoidal function with a period of 2π, an amplitude of 1, and passing through the point (-π/2, 1). Learn the mathematical steps involved in determining the phase shift and constructing the function. Suitable for high school students studying trigonometry and sinusoidal functions.

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