consequently, all distances estimated in a direction away from the extremity of the arc must be considered negative.

Thus, OT, regarded as the secant of AM, is estimated in a direction toward M, and is positive; but OT, regarded as the secant of AM", is estimated in a direction away from M", and is negative.

These conventional rules enable us to give at once the proper sign to any function of an arc in any quadrant.

59. In accordance with the above rules, and the definitions of the circular functions, we have the following principles:

The sine is positive in the first and second quadrants, and negative in the third and fourth.

The cosine is positive in the first and fourth quadrants, and negative in the second and third.

The versed-sine and the co-versed-sine are always positive.

The tangent and cotangent are positive in the first and third quadrants, and negative in the second and fourth.

The secant is positive in the first and fourth quadrants, and negative in the second and third.

The cosecant is positive in the first and second quadrants, and negative in the third and fourth.

LIMITING VALUES OF THE CIRCULAR FUNCTIONS.

60. The limiting values of the circular functions are those values which they have at the beginning and the end of the different quadrants. Their numerical values are discovered by following them as the are increases from 0° around to 360°, and so on around through 450°,

540°, &c. The signs of these values are determined by the principle, that the sign of a varying magnitude up to the limit, is the sign at the limit. For illustration, let us examine the limiting values of the sine and the tangent.

If we suppose the arc to be 0, the sine will be 0; as the arc increases, the sine increases until the arc becomes equal to 90°, when the sine becomes equal to +1, which is its greatest possible value; as the arc increases from 90°, the sine diminishes until the arc becomes equal to 180°, when the sine becomes equal to +0; as the arc increases from 180°, the sine becomes negative, and increases numerically, but decreases algebraically, until the arc becomes equal to 270°, when the sine becomes equal to - 1, which is its least algebraical value; as the arc increases from 270°, the sine decreases numerically, but increases algebraically, until the arc becomes 360°, when the sine. becomes equal to 0. It is 0, for this value of the arc, in accordance with the principle of limits.

The tangent is 0 when the arc is 0, and increases till the arc becomes 90°, when the tangent is; in passing through 90°, the tangent changes from tox, and as the arc increases the tangent decreases numerically, but increases algebraically, till the arc becomes equal to 180°, when the tangent becomes equal to - 0; from 180° to 270° the tangent is again positive, and at 270° it becomes equal to +; from 270° to 360°, the tangent is again negative, and at 360° it becomes equal to 0.

-

If we still suppose the arc to increase after reaching 360°, the functions will again go through the same changes, that is, the functions of an arc are the same as the functions of that are increased by 360°, 720°, &c.

By discussing the limiting values of all the circular functions we may form the following table:

61. Let AM, denoted by a, represent any are whose radius is 1. Draw the lines as represented in the figure. Then we shall have,

From the right-angled triangle OPM, we have,

The symbols sin a, cos a, &c., denote the square of the

sine of a, the square of the cosine of a, &c.

From formula (1) we have, by transposition,

From the similar triangles OAT and OPM, we have,

(5.)

From the similar triangles ONM and OBT', we have,

Multiplying (6) and (7), member by member, we have,

From the similar triangles OPM and OAT, we have,

From the similar triangles ONM and OBT', we have,

From the right-angled triangle OAT, we have,

From the right-angled triangle OBT', we have,

It is to be observed that formulas (5), (7), (12), and (14), may be deduced from formulas (4), (6), (11), and (13), by substituting 90°-a, for a, and then making the proper reductions.

Collecting the preceding formulas, we have the following table: