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 cos2a 1 sin2a.

(3.)

R

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,
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,

ŪB2 + BT"2; or, co-sec2a = 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.

FUNCTIONS OF NEGATIVE ARCS.

62. Let AM"", estimated from A towards numerically equal to AM; then,

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° + ɑ.

We see from the figure that,

By a simple inspection of the figure, observing the rul for signs, we deduce the following relations :

without reference to their signs: hence, we have, as before,

By a similar process, we may discuss the remaining arcs Collecting the results, we have the following

in question.

table: