SEMANTICS OF THE DIFFERENTIAL CALCULUS 599 

The following table gives the ordinary and radian measures, the sine, cosine and tangent of angles of 0, 1, 2, 3, and 4 right angles. 







From Fig. 14 and from this table, it follows that the values of the trigonometric functions are equal for the angles 0 and 2ir, or in the language of degrees, for the angles 0° and 360°. We see also from Fig. 14 that the angle XOP\, or any other angle, has one measure as expressed by its trigonometric functions if we add to it 360° or 2ir radians.
The structural importance of the trigonometric functions in analysis lies in the fact that they are the simplest singly periodic functions and are therefore adapted for the representation of undulations. As we have already seen the sine and cosine have the single real period 2, which means that they are not altered in value by the addition of 2ir to the variable. The tangent has the period.
Besides the three functions defined above, we usually define three others, the secant, the cosecant and the cotangent as reciprocals respectively of the cosine, the sine, and the tangent. These last three we may disregard in our present discussion.
Let us consider the function y = sin x, and construct the curve which this equation represents. If we draw a circle of unit radius, Fig. 14, the ordinates corresponding to the different angles XOPi, XOPt., give the values of y, while the angles measured in radians, give the corresponding values of the abscissa x.
Plotting corresponding values of * and y as thus obtained in Fig. 14 we get in Fig. IS the partial graph of the function y = sin x. Proceeding again around our circle in Fig. 14, that is, adding 360° or 2ir, to each of our angles, hence to their abscissas of the curve in Fig. IS, we add to the graph a second complete wave. We may thus proceed either forward or backward obtaining as many complete waves, or undulations, as we please, as in Fig. 16.
The curve represented by y = cos x is obtained in like manner and is quite similar to the sine curve. (See Fig. 17.)

