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Sinusoids: Centerline, Amplitude, Phase Angle & Period

Instructor: Michael Eckert

Michael has a Bachelor's in Environmental Chemistry and Integrative Science. He has extensive experience in working with college academic support services as an instructor of mathematics, physics, chemistry and biology.

Sinusoid functions, e.g. sine or cosine, have specific characteristics such as an amplitude, period, phase angle and a centerline. In this lesson, we seek to address and illustrate each one of these features or characteristics.

Characteristics of Sinusoids: Amplitude, Period, Phase Angle and Centerline

Given a sine function, whether y = sinx or y = cosx, certain characteristics common to either type of sinusoid are present. To begin to address these characteristics, we can rewrite y = sinx as y = Asin(Bx + C) or y = cosx as y = Acos(Bx + C) respectively, where Amplitude = A, Period = 2π / B and Phase Angle = C. Note: Centerline will be dependant on vertical position and amplitude of the sinusoid.

y =sinx or y = Asinx(Bx + C)

In looking at the graph of the sine function y = sinx below, we can go over the concepts of said characteristics i.e. Amplitude (A), Period (2π / B) and Phase Angle (C):


sin


Amplitude

Starting from the origin (0 , 0), we see this function increase at it reaches a maximum at x = 1.57 (π / 2) or 90 degrees at y =1:


sma


We see it decrease through y = 0 again before reaching its minimum at y = -1 at (4.71, -1) or (3π / 2 , -1)


smi


The amplitude (A) of our function A = 1 is the absolute value of either our minimum or maximum y-value. The amplitude of this function can be enlarged by merely increasing the value of A. For instance, if we multiply the value of A by 2, our amplitude increases by a multiple of 2, as shown in the graph of y = 2sinx superimposed onto y = sinx:


amp


Period

For our purpose here, the period of our sine function y = sinx is simply represented by 2π / B. In this case, B is just 1, so the period is just 2π or 6.28. 2π is another way of saying 360 degrees. Period is that value through which our function makes 1 complete cycle.


period


We see our function y = sinx start at the origin, reach a maximum at x = π / 2 , descend back through the origin at π to its minimum at x = 3π / 2, then increase back up to 2π. And note that just as our amplitude can be stretched, so can our period. For instance, suppose we wanted to increase our period by a factor of 2. In this case B = ½, and since period is proportional to 2π / B, the period will be multiplied by the reciprocall of ½, which is 2. In this case, our period will increase to 4π or 720 degrees.

Centerline

Centerline is simply the midway point along the vertical distance of y between the maximum and minimum y-values. In other words, if y = A sinx and amplitude equals 1 and the maximum y-value is 1 and the minimum y-value is negative 1, then the centerline would be at y = 0.

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