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):



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:


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


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:



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.


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 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.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account