Write an equation and for each sine function described. Each cycle behind at the origin.

a) amplitude=6, period=180

b) amplitude=2, period=540

Begins*

To write an equation for each sine function, we need to understand the general form of a sine function:

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

Where:
A = amplitude
B = 2π / period
C = phase shift
D = vertical shift

Let's proceed with each sine function:

a) For amplitude = 6 and period = 180, we have:
A = 6
Period = 180 degrees

First, we need to convert the period from degrees to radians. Since there are 360 degrees in a circle, to convert to radians, we multiply by π / 180:

Period in radians = 180 * π / 180 = π

Next, we can rearrange the formula to solve for B:

B = 2π / Period in radians = 2π / π = 2

Now, since the cycle begins at the origin (phase shift = 0) and there is no vertical shift (D = 0), the equation becomes:

y = 6 * sin(2x)

b) For amplitude = 2 and period = 540, we have:
A = 2
Period = 540 degrees

Again, we convert the period from degrees to radians:

Period in radians = 540 * π / 180 = 3π

Now, we can solve for B:

B = 2π / Period in radians = 2π / (3π) = 2/3

Since the cycle begins at the origin (phase shift = 0) and there is no vertical shift (D = 0), the equation becomes:

y = 2 * sin((2/3)x)

Therefore, the equations for the sine functions are:

a) y = 6 * sin(2x)
b) y = 2 * sin((2/3)x)