Last-Minute Comments Before Seeing The Pictures

Two comments are in order before we actually plot these graphs.

First, note that the trig functions and are completely determined for ALL values of once you know and for . This is because a circle spans 360° in one revolution. Thus, for example, the terminal side of the angle 405° lies in exactly the same place as the terminal side of the angle 45° because 405 = 45 + 360. So, sin 405° = sin 45°. This is also true, for example, for the angles 1400° and 320°. Therefore, if one wishes to calculate the trig value of an angle, one can ``subtract out'' all multiples of 360 from the angle and then study the remaining angle, which is an angle between 0° and 360°. (The same can be said for negative angles, except that one ``adds in'' multiples of 360 rather than subtracting out multiples of 360.)

Because of this, the graphs of and will repeat themselves every 360° or radians. Indeed, once we have the graphs for x between 0 radians and radians, we can simply copy this portion forever in both directions to get the complete graph.

Second, note that, in the case of and , it will always be the case that and . This can be seen in a variety of ways. The most important item to realize here is that and where x and y are the lengths of legs of the right triangle in question and r is the length of the hypotenuse. From the Pythagorean Theorem x² + y² = r² , it should be clear that x and y can never be bigger than r. From a visual point of view, the hypotenuse of a triangle can never be shorter than either of the two legs. (You would not be able to connect the three legs if it were the case that the hypotenuse was shorter than one of the legs.) Hence, we must have and .

Because of this fact, we can see that the graphs of and must always lie between -1 and 1 in the y-direction. Using a more technical term, we say that the range of the functions is the interval [-1, 1].