The work of a curious fellow
There is always and angle isn't there...

In our discussion of numbers, functions and graphs , we spoke of a function as a rule relating an input set of numbers to an output set. If you need to review the concept you may jump to the link above. Now, in particular, we will look at some specific functions that arise from the study of the relationship between lines and angles known as trigonometry.

Rather than attempting a general treatment of the topic, we will focus our attention on just three of the several trigonometric functions. These functions are named the sine function, the cosine function and the tangent function. To get at the definition of the functions we have to begin by defining some more fundamental terms.

A triangle is a three sided figure formed by the intersection of three non-parallel straight lines, like this.
If one of the angles in the triangle is a right angle (an angle of 90 degrees) like this,
Right Triangle
the triangle is called a right triangle. The relationships among the sides and angles of a right triangle is the basis for the sine and cosine functions.

Let's label the parts of the right triangle. The side opposite the right angle is called the hypotenuse. (Don't ask why.) We will call one of the acute (less than 90 degree) angles "A" and the other "B". The side opposite "A" will be labeled "a" and that opposite "B", "b". Like this.
Labeled Triangle

Now we have the elements needed to define the functions. The sine of the angle "A", written as sin(A), is the fraction "a" divided by the hypotenuse.


The cosine of the angle "A", written as cos(A), is the fraction "b" divided by the hypotenuse.

The tangent of the angle "A", written as tan(A), is the fraction "a" divided by "b".

Since "a" is the side of the triangle opposite to the angle "A", the sine is sometimes thought of as opposite over hypotenuse. The cosine in similar terms is adjacent over hypotenuse and the tangent is opposite over adjacent. As an exercise, define for yourself the sine, cosine and tangent of the angle "B".