A sinusoidal function has an amplitude of 3 units, a period of 180 degrees and a max point at (0, 5). Represent the function with an equation in two different ways.
First equation:
A general equation for a sinusoidal function is given by:
y = A*sin(B(x-C)) + D
Where:
A = amplitude
B = 2π/period
C = phase shift (horizontal shift)
D = vertical shift
Given that the amplitude is 3 units and the period is 180 degrees, we can substitute these values into the equation:
y = 3*sin((2π/180)(x-C)) + D
Since the max point is at (0, 5), we substitute the x and y values into the equation to find C and D:
5 = 3*sin((2π/180)(0-C)) + D
5 = 3*sin(-2πC/180) + D
Since the maximum value is positive, sin(-2πC/180) must equal 1:
5 = 3*1 + D
5 = 3 + D
D = 2
So, the first equation becomes:
y = 3*sin((2π/180)(x-C)) + 2
Second equation:
Another way to represent a sinusoidal function is using cosine instead of sine. The general equation for a cosine function is:
y = A*cos(B(x-C)) + D
Since sine and cosine differ only by a phase shift of π/2 radians, we can rewrite the equation as:
y = 3*cos((2π/180)(x-C-90)) + 2
Therefore, the second equation is:
y = 3*cos((2π/180)(x-C-90)) + 2