Using Trigonometry




Modifying Curves

Adding Curves



Dido's Problem



Smart Joe

Fuzzy Logic



While in the 16th century the Italian mathematician Galileo was studying objects in motion, he did not only discover the basic laws about motion, but he also invented the idea of the function which simplifies calculations in mathematics and physics.

In contrary to the Greeks, Galileo was convinced of the importance of experiments and studied carefully their results. He didn't try to explain nature, but to describe it. And to do this, functions are a great instrument.

Galileo found for example that any object which is dropped will accelerate by 16 feet per square second if friction from air is neglected. This means that every second an object will get faster 16 feet per second. The distance that an object falls within a given time when dropped he found to be equal to the number of seconds squared times 16 feet. (That is, of course, only as long as the object doesn't hit the ground.)

We could describe this with the following equation:


Lets look at what this equation tells us. In this case, d is the distance and t is the number of seconds the object falls. The neat thing about this equation is that now you can easily find the distance an object travels in any given time: we can find d for any t. Note that for every t there is only one d. Any equation which satisfies the above two statements is a function.

Functions today are usually written in the way devised by Euler. After Euler the above function is written as follows:


The part on the left of the equal sign is the name of the function f, and the t inside the () denotes the argument. When you use the function, for example to find the distance a ball falls in two seconds, you write f(2). To find the value of f(2) you set s equal to 2 and then find the value of the right side of the function. For example, in this case replace the t²×16 by 2²×16=4×16 which is equal to 64. So the ball falls 64 feet.

Lets go back now to the trigonometric ratios. They can be defined as functions of an angle. For example, take the sine. For every angle there is exactly one coresponding ratio, in the case of the sine this is the side in a right triangle which is oposite the given angle divided by the near angle. Since both of these sides have exactly one length and a division of two numbers results in just one number the sine of an angle satisfies all restrictions placed upon functions. So sin(30°) is defined as the sin for an angle of 30°. The same goes for the other trigonometric ratios.