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 goes on diminishing 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 goes on increasing numerically, but decreasing 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 goes on decreasing numerically, but increasing algebraically, until the arc becomes 360°, when the sine becomes equal to -0. It is 10, for this value of the arc, in accordance with the principle of limits.

2,

and

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 + လ to as the arc increases the tangent decreases, numerically, but increases algebraically, till the arc becomes equal to 180°, when the tangent becomes equal to 270°, the tangent is again positive, equal to +; from 270° to 360°, the negative, and at 360° it becomes equal to

0; from 180° to and at 270° it becomes

tangent is again

· 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 that are increased by 360°, 720° &c.

By discussing the limiting values of all the circular functions we are enabled to form the following table :

any arc de

shall

M

N

61. Let AM represent noted by a.. Draw the lines as represented in the figure. Then we have, by definition

From the right-angled triangle OPM, we have,

The symbols sin2a, cos2a, &c., denote the square of the sine of a, the square of the cosine of a, &c. From Formula (1) we have, by transposition,

sin2a = 1 cos2a (2); and cos1a = 1 — sin2a.

(3.)

From the similar triangles OAT and OPM, we have,

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

ON: NM:: OB: BT', or, sin a-: cosa:: 1: cot a;

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,

ON: OM :: OB : OT, or, sin a : 1:: 1: co-sec a;

From the right-angled triangle OAT, we have,

(12.)

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

OTA

= OB2 + BT2; or, co-seca = 1 + cot2a. . (14.)

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,

ure, we shall discover the following relations, viz.:

FUNCTIONS OF ARCS FORMED BY ADDING AN ARC TO, OR SUBTRACTING IT FROM ANY NUMBER OF QUADRANTS.

63. Let α denote any arc less than 90°.

has preceded, we know that,

From what

Now, suppose that BM' = a, then will AM' = 90° + a.

We see from the figure that,