In this lesson we will investigate the begaviour of the trigonometric functions.
> plot(sin(x), x = -2*Pi..2*Pi);
> plot(cos(x), x = -2*Pi..2*Pi);

Plot tan, cot, sec and csc on the interval [-2Pi, 2Pi]. (Note that these functions all have vertical asymptotes.)
You should know these graphs well.

The trigonometeric functions are all periodic, that is they repeat thier form at regular intervals. sin, cos, sec and csc have period 2Pi, whereas tan and cot have period Pi. sin and cos are bounded between -1 and 1, whereas the others all have infinite discontinuities.

sine and cosine are particularly important in that they can be used to model waves and wavelike phenomena.

Amplitudes

As we might expect from the last lab the effect is to stretch the graph of sine vertically. This raises the hieght of the peaks and troughs of sine, this is called the amplitude.

What happens when A is not a constant?
> A := x -> x^2;
> plot([A(x), A(x)*sin(x)], x = -2*Pi..2*Pi);

It seems that A(x) effects the amplitude at each point by the amount of its value at that point. To see this more clearly lets extend the range:
> plot([A(x), -A(x), A(x)*sin(x)], x = -10*Pi..10*Pi, color=[red,green,blue]);

The effect for the other trigonometric functions is much the same.

1. Plot A(x), -A(x) and A(x) sin(x) on the same graph for the following functions A(x):
   • A(x) = x.
   • A(x) = |x|.
   • A(x) = 1/x.
   • A(x) = 1/x^2.
2. Repeat the above exercise for cos.
3. Repeat the above exercise for tan.
4. Repeat the above exercise for sec.

Multiplying Trigonometric Functions Together

This looks alot like tan. The reason for this is not immeadiatly obvious from the graph, however algebraicaly it is obvious:
sec(x)sin(x) = sin(x)/cos(x) = tan(x), since sec(x) is 1/cos(x).

Many products of trigonometric functions can be understood by applying trigonometric identities.
For example consider sin(x)cos(x) = ½ sin(2x)
> plot([sin(x), cos(x),cos(x)*sin(x)], x = -3*Pi..3*Pi, color=[red,green,blue]);

In this context the Maple function combine can be useful. For example try
> combine(cos(x)*sin(x));

5. Plot the following, and explain the resulting graphs.
   1. sin²x
   2. cos²x
   3. 4 cos²x - 1.

Frequency

We can see that sin(2x) wiggles twice as fast, that is it has ½ the period of sin(x). In general sin(nx) will wiggle n times as fast as sin(x). This works for other trigonometric functions as well.

Note that the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b) allows us to find expansions for these functions.
> expand(sin(2*x));
We can do a whole range:
> t := [ sin(n*x) $ n = 2..10];
> s: = expand(t);
Perhaps more importantly, the combine function allows us to combine these back.
> combine(s);

Note that the functions sin(nx) and cos(nx), where n is an integer, are also very important mathematically. You will meet these functions again if you do fourier series.

6. On the same graph plot sin(nx) for n = 1,2,3,4,5, on the range -Pi to Pi.
7. On the same graph plot sec(x) and sec(2x).
8. On the same graph plot sec(x) and sec(3x).
9. On the same graph plot tan(x) and tan(2x).
10. On the same graph plot tan(x) and tan(3x).

Phase

Once agian the trig identities can be used to expand this sort of form:
> s := expand(sin(x + d));

And we can combine these back together:
> combine(s);

Wave Functions

A is the Amplitude of the wave.
w is the freqency of the wave.
d is the phase of the wave.

We could equally well have used a cosine function, but since cos(x) = sin(x + Pi/2) we can use the phase to cover cosine waves.

11. Plot the following wave functions, on the same graph plot the amplitudes.
    1. A = 5, w = 5, d = 0.
    2. A(x) = 1/x, w = 15, d = Pi/4.
    3. A(x) = exp(-x²), w = 15, d = 0.
    4. A(x) = sin(x), w = 15, d = 0.

Summing Trigonometric Functions

This shows that the higher frequency wave just oscillates using the other as a baseline. In fact this sort of behaviour will happen for other functions as well.
> plot([sin(11*x), x^2, x^2+sin(11*x)], x = -1.5*Pi..1.5*Pi, color=[red,green,blue]);

We can us this idea to build up an idea of what functions might look like, having oscillations within oscilations.
> plot([sin(x), sin(x) + sin(3*x)], x = -2*Pi..2*Pi, color=[red,blue]);
> plot([sin(x) + sin(3*x), sin(x) + sin(3*x) + sin(12*x)], x = -1.5*Pi..1.5*Pi, color=[red,blue]);

We can emphasise the effect of a particular cycle by giving it a higher amplitude.
> plot([sin(x) + 3*sin(3*x), sin(x) + 3*sin(3*x) + sin(12*x)], x = -1.5*Pi..1.5*Pi, color=[red,blue]);

12. 1. On the same graph plot sin(x), sin(2*x) and sin(x)-3*sin(2*x)
    2. On the same graph plot sec(x) and sec(2x) and sec(x) + sec(2x).
    3. On the same graph plot tan(x) and tan(3x) and tan(x) + tan(3x).
    4. Explain the results of these plots.

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