Using Trigonometry




Modifying Curves

Adding Curves



Dido's Problem



Smart Joe

Fuzzy Logic



The most interesting part of graphing trigonometric functions is to scale and move the curve by changing the equation. On the previous page, the sine function was graphed as y=sin(x). By adding some additional constants to that equation, the curve can be stretched and moved. This is particularly useful when you have a sine curve to model a physical process, for example a sound wave.

You can click onto the numbers below to scale and move the curve. Just press your mouse button on top of one of the numbers in the top equation and then drag up or down. You will see the equation and the curve updating immediately. (Note: the numbers are rounded to one digit behind the decimal point.)

In the second equation, the numbers are replaced with the letters a-d. You'll find a short explanation of physical meaning of these letters. You may even hear the soundwave if you click on play. The sound will automaticaly reflect any changes you make to the wave. (Not all changes, however, can be heard.)

Can you move the curve up and down?

How do you increase the amplitude? (The amplitude is the difference between the highest and the lowest point of the curve.) Does the curve sound different when the amplitude changes?

Now change the period of the wave. (The period is the lengh it takes for the wave to repeat itself. If you want to find the period of the wave, measure the distance from one of the highest points over to the next point which has the same height.)

When working with sound it is usually easier to work with the frequency of a wave instead of its period. (Frequency is the number of times the curve goes up and down in a given time and is usually measured in hertz. A wave of one hertz goes up and down once per second. A wave of two hertz does the same twice a second. A sound needs a minimum frequency of about sixty hertz to be auditable. The frequency is inverse proportional to the length of a wave. This is actually quite obvious. If you have a wave of one hertz it goes up and down once a second and it needs one second to go up and down. Now take a wave of two hertz. This wave goes up and down twice a second but only takes half a second to go up and down.) The higher the frequency of a wave, the smaller the period of the wave. Compare the sound of waves whose period differ by a whole multiple of one. For example, compare a wave with another one which has half the period of the first one.