What is the equation of a sine curve that has an amplitude of 2 and a period of 4π radians?

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To find the equation of a sine curve, we can use the general formula for the sine function:

y = A * sin(B(x - C)) + D

Where:

  • A is the amplitude of the sine wave, which determines its maximum height.
  • B affects the period of the wave, which is the distance over which the wave repeats.
  • C is the horizontal shift (phase shift) of the sine curve.
  • D is the vertical shift of the sine curve.

In your case:

  • The amplitude A is given as 2.
  • The period is given as 4π radians.

The period P of a sine function is calculated with the formula:

P = 2π / B

To find the value of B, we set the period equal to 4π:

4π = 2π / B

Rearranging this equation gives:

B = 2π / 4π = 1/2

Now we have:

  • A = 2
  • B = 1/2
  • C = 0 (since there’s no horizontal shift assumed here)
  • D = 0 (for no vertical shift)

With these values, we can write the equation for the sine curve:

y = 2 * sin((1/2)x)

This equation describes a sine curve with an amplitude of 2 and a period of 4π radians.

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