2003-11-14 11:44:56 -- The six commonly used trigonometric functions are defined as follows:
Since x and y do not change if 2p radians are added to the angle—that is, 360° are added—it is clear that sin (è 2p) = sin è. Similar statements hold for the five other functions. By definition, three of these functions are reciprocals of the three others, that is,
If point P, in the definition of the general trigonometric function, is on the y-axis, x is 0; therefore, because division by zero is inadmissible in mathematics, the tangent and secant of such angles as 90°, 270°, and -270° do not exist. If P is on the x-axis, y is 0; in this case, the cotangent and cosecant of such angles as 0°, 180°, and -180° do not exist. All angles have sines and cosines, because r is never equal to 0.
Since r is greater than or equal to x or y, the values of sin è and cos è range from -1 to 1; tan è and cot è are unlimited, assuming any real value; sec è and csc è may be either equal to or greater than 1, or equal to or less than -1.
It is readily shown that the value of a trigonometric function of an angle does not depend on the particular choice of point P, provided that it is on the terminal side of the angle, because the ratios depend only on the size of the angle, not on where the point P is located on the side of the angle.