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Recognizing & Modeling Periodic Functions

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.

Trigonometric functions such as sine and cosine are periodic in nature, exhibiting repeating waves through a given period. These functions serve to model physical phenomena such as sound waves and / or uniform circular motion.

Recognizing Trigonometric Periodic Functions

Sine and Cosine

Trigonometric functions—specifically the sine and cosine functiions—are often used to model periodic phenomena over a variety of scientific disciplines. As functions, they exhibit key characteristics necessary for modeling such phenomena, e.g. sound and motion. We can see this graphically. They have wavelength, amplitude, and period. Therefore, we can see how the sine (y = sinx) and cosine (y = cosx) might be implemented in modeling for harmonic and uniform circular motion in particular.

Within the Cartesian coordinate system, y = sinx and y = cosx are periodic in that they have repeating y- values over an interval of time. In both y = sinx and y = cosx, y will have a corresponding frequency and amplitude and a reoccurring period through 360 degrees (or 2Π radians). Each y -value has an angular position through a period corresponding to the trigonometric unit circle.


sine wave


Note that period is the time through which a wave completes one cycle; whereas frequency is a measure of how many waves cycle through a given time. Amplitude is merely the height of the peaks of the wave.

Sine and Cosine Graph

The graph for y = sinx for instance is simply a wave with repeating peaks and valleys through 360 degrees (in this case 0-10 on x-axis):


sine


When the graph for y = cosx is superimposed onto y = sinx, they become easy to compare:


sine and cos


y = cos x appears merely as y = sinx shifted to the left or right -remembering that 1-10 on x-axis is 360 degrees.

Physics Modeling with Sine and Cosine

Harmonics

In harmonics, or the study of sound, these functions can be used for modeling. Note that a sound wave exhibits an amplitude (A), an angular frequency (ω) and a period (θ). Let's take y = Asin(ωx + θ) for instance, where A = 2 and ω = 60 degrees/sec or the speed at which said wave is passing through its period θ = 360 degrees. If we graph y = 2sinx (60x + 360):


sine2


When y = Acos(ωx + θ) is superimposed onto y =Asin(ωx + θ), we get:


sincose2


Note that when we get a sine and cosine representing two waves of sound that are reflected across the x-axis, we get something called a standing wave. This is where the term harmonic is derived -as in the 1st, 2nd, and 3rd harmonic for example.


har


1st, 2nd, and 3rd harmonic is determined by the number of intersection points of the two waves or nodes:


node


Uniform Circular Motion

As in harmonics, the sine and cosine function may also be used in the analysis of motion (specifically uniform circular motion). A way to conceptualize this connection between the trigonometric function and circular motion might be to take look at a unit circle


uc


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