The functions of angles can best be illustrated 0 along the x axis to thc right are 0 along the y axis from 0 upward are positive; coordinates measured along the y axis from 0 downward are negative.

Angles are generated by the motion of a point P counterclockwise along the circumference of the circle. The initial leg of any angle is the positive leg of the x axis. The other leg is the radius r, at the end of which the point P is located; this radius always has a value of 1. The unit radius (r = OC) is subdivided into 10 equal parts, so the value of each of the 10 subdivisions shown is 0.1.  

For any angle, the point P has three coordinates: the x coordinate, the y coordinate, and the r coordinate (which always has a value of 1 in this case). The functions of any angle are, collectively, various ratios that prevail between these coordinates.

The ratio between y and r (that is, y/r) is called the sine of an angle. In figure 1-21, AP seems to measure about 0.7 of y; therefore, the sine f, which is equal to 45 in this case, would seem to be 0.7/1, or about 0.7. Actually, the sine of 45 is 0.70711. Graphically, the sine is indicated in figure 1-21 by the line AP, which measures 0.7 to the scale of the drawing.

The ratio between x and r (that is, x/r) is called the cosine of the angle. You can see that for 45, x and y are equal, and the fact that they are can be proven geometrically. Therefore, the cosine of 45 is the same as the sine of 45, or 0.70711. Graphically, the length of line OA represents the cosine of angle f when the radius (r) is equal to 1.

The ratio between y and x (that is, y/x) is known as the tangent of an angle. Since y and x 1-21, you can deduce that BC is equal to OC. OC is equal to the unit radius, r.

The three functions shown in figure 1-21 are called the "direct" functions. For each direct function there is a corresponding "reciprocal" functionmeaning a function that results when you divide 1 by the direct function. You know that the reciprocal of any fraction is simply the cosecant) is divided by y/r, which is r/y.

For the direct function cosine, which is x/r, the reciprocal function (called the secant) is r/x. Since x for cosine 45 also measures about 0.7, it follows that the secant for 45, r/x, is the same as the cosecant, or also about 1.4. The secant is indicated graphically in figure 1-21 by the line OB also.

For the direct function tangent, which is y/x the reciprocal function (called the cotangent) is x/y. Since x and y at tangent 45 are equal, it follows that the value for cotangent 45 is the same as that for the tangent, or 1. The cotangent is shown graphically in figure 1-21 by the line DB, drawn tangent to the circle at D.