Trigonometric Functions

THERE IS ALWAYS AN ANGLE...

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,

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.

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".

The sine and cosine functions are closely related to circular motion. Consider what values the sine and cosine take on as the angle "A" sweeps around in a complete circle. The display below illustrates that effect. Click on Action to start the angle increasing and Cut to halt it. Notice that the units on the angle measure is in radians .

It is clear that whatever the angle "A", the sine of "A" and cosine of "A" will always be less than or equal to 1 and greater than or equal to -1. So the sine and cosine as defined conform to our idea of a function, taking a real number, "A", and returning real numbers in the range -1 to 1.

Since our angle is a function of time in this display, we can plot the sine and cosine as functions of time. In the next display we will see the rotating vector, marking the angle as before and the corresponding plot of the sine and cosine functions.

These two functions show up frequently in mechanics because, as you can see, they relate rotation to a waving kind of motion. Many interesting systems in mechanics exhibit one or the other of these sorts of motion.

Notice that the sine and cosine functions differ only in their starting time. The wave shapes are identical but the sine lags the cosine by 1/4 cycle or 90 degrees of rotation.

Use the browser "Back" feature to return from whence you came.