6. Composite Trigonometric Curves
by M. Bourne
When adding 2 or more trigonometric graphs, we add ordinates (y-values) to find the resultant curve.
Example 1: Electronics
interactive spring activity
One of the simplest examples of composite trigonometric curves is from electronics. An AC (alternating current) signal (v = 3 cos 8t) is added to a `10` volt DC (direct current) voltage source. The resulting signal looks like the following.
The combined graph that you see has equation: v = 10 + 3 cos 8t.
We are simply adding `10` to all the v-values of the cosine curve.
When two water waves meet on a pond, they combine such that when 2 crests meet, they are added to give a larger crest, and when 2 troughs meet, they add to give a deeper trough. A crest and a trough tend to cancel each other out when they meet.
In this example, let's assume the following 2 waves meet.
`a(x) = 5\ sin\ x`
`b(x) = 4\ cos(2x + π/3) `
Need Graph Paper?
Sketch the graph of the combined wave:
`y=5\ sin(x)+4\ cos(2x+pi/3)`
Example 3: Double Springs Flash Interactive
We have two springs with different sized masses connected and hanging vertically. While holding the top mass still, we pull down the bottom mass. Then we let go of both masses and allow the system to move freely.
A motion sensor is connected to a computer and we can see the resulting movement of the masses as time progresses.
In this Flash animation, you can reset the motion at any time and get a different set of resulting graphs. Just click on the round button at the bottom.
- This model is not realistic, of course, since our springs never slow down. Also, they are constrained to move in one dimension only.
- You can see other interesting spring examples in the previous section, Applications of Trigonometric Curves and in the section on Work, which is an application of Calculus.
- You cannot grab the masses in this example - it resulted in motion that was too chaotic.
Depending on the masses, the lengths of the springs and the spring constants, we could get a curve similar to the following, which is the sum of two cosine curves:
x = 0.0572 cos(4.667t) + 0.0218 cos(12.22t)
Reference: Morland, T "Modeling a Simple Mechanical System", Teaching Mathematics and Its Applications, Vol 18 No 2 1999.
1. Graph the composite trigonometric curve `y = x^2/10 − sin\ πx`
2. Graph the curve y = 2 cos 2x + 3 sin x
Example 4: Real-world Case
Atmospheric carbon dioxide levels have been increasing since the beginning of the industrial revolution.
This chart from the National Oceanographic and Atmospheric Administration shows the increase in CO2 since records began in 1958, on Mauna Loa in Hawaii.
We can closely model this curve as the sum of a cosine and cubic curve as follows:
y = 3.07 cos(2πx − 1.2) + 0.00002052590807(x − 1958)3
+ 0.01105601542(x − 1958)2 + 0.8044611048(x − 1958)
Here is the graph of the model. It is similar to Exercise 1 above.
For more background and information on this model, see Earth killer - composite trigonometry CO2 graph.
Composite Trigonometric Graph - Product of Functions
The following examples show composite trigonometric graphs where we are taking the product of two functions.
Example 5: y = x sin x
In this example, we are multiplying the sine of each x-value by the x-value.
So for example, if `x = 2`, the y-value will be `y = 2\ sin\ 2 = 1.819`.
Example 6: RCL Circuit - Constant Forced Response
In electronics, an RCL circuit has a resistor, a capacitor and an inductor connected in series. If we apply a constant voltage to the circuit, there is an initial pulse in the current. The resulting function is a product of an exponential decay function and a sin function.
i = e-0.3t sin t
The graph of the current at time t has the following appearance. The current reduces quickly after the initial pulse.
We learn how to solve these problems later in Applications of Second Order Differential Equations.