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 nonparallel straight lines, like this.
If one of the angles in the triangle is a right angle (an angle
of 90 degrees) like this,
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.
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.
sin(A)=a/hypotenuse
The cosine of the angle "A", written as cos(A), is
the fraction "b" divided by the hypotenuse.
cos(A)=b/hypotenuse
The tangent of the angle "A", written as tan(A), is
the fraction "a" divided by "b".
tan(A)=a/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".
