6. Composite Trigonometric Curves

by M. Bourne

When adding 2 or more trigonometric graphs, we add ordinates (y-values) to find the resultant curve.

Later, on this page...

Spring with 2 masses giving composite trigo graph

See the interactive spring activity below for an example of composite trigonometric curves.

Example 1: Electronics

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.

Example 2

Ocean waves
Water waves

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) `

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Sketch the graph of the combined wave:

`y=5 sin(x)+4 cos(2x+pi/3)`

Continues below

Example 3: Double springs interactive graph

We have two springs with different thicknesses (and spring constants) with two 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.

We can also grab the mass in the middle to compress or stretch the top spring.

In a real experiment, we'd have a motion sensor connected to a computer and we would be able to see the resulting movement of the masses as time progresses, just as you can in this simulation.

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Stretch (or compress) either of the springs by dragging either of the masses, then let go to see the resulting waveforms. Such waveforms are the result of adding different trigonometric curves.

Notes:

  1. Our springs are constrained to move in one dimension only.
  2. The springs slow down as time goes on, due to friction.
  3. You can see other spring examples in Graphs of y = a sin bx and y = a cos bx, and in the section on Work, which is an application of Calculus.

Depending on the masses, the lengths of the springs and the spring constants, we could get a curve similar to the following (for one or the other mass positions), which is the sum of two cosine curves:

x = 0.0572 cos(4.667t) + 0.0218 cos(12.22t)

[The function x(t) given above is obtained using differential equations, an interesting topic which we meet later in the calculus section.]

Reference: Morland, T "Modeling a Simple Mechanical System", Teaching Mathematics and Its Applications, Vol 18 No 2 1999.

Exercises

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.

CO2 concentration increase 1960 to 2016

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) + 314.634017

Here is the graph of the model. It is similar to Exercise 1 above.

CO2 concentration increase

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.

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