# 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

One of the simplest examples of composite trigonometric curves is from electronics. An AC (alternating current) signal (*v* = 3 cos 8*t*) 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 8*t*.

We are simply adding `10` to all the *v*-values of the cosine curve.

### Example 2

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) `

### Need Graph Paper?

Sketch the graph of the combined wave:

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

### Example 3: Double Springs

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.

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)

Graph of `x=0.0572 cos(4.667t) + 0.0218 cos(12.22t)`, a composite trigonometric model of a double spring system.

[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 2*x *+ 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)

+ 314.634017

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}sint

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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