From the similar triangles OAT and OPM, we have,

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,

=

OT" OB2 + BT2; or, co-sec2a = 1 + cot❜a.

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, a, for for a, and then making the proper reductions.

FUNCTIONS OF NEGATIVE ARCS.

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

if we denote the arc AM by a, the arc AM"" will be denoted by - a (Art. 48).

All the functions of AM"", will be the same as those of ABM""; that is, the functions of

T"

T

M'

IN

MT

tions of 360° a.

From an inspection of the fig

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

M

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

63. Let a denote any arc less than 90°. has preceded, we know that,

From what

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 in question. Collecting the results, we have the following table:

It will be observed that, when the arc is added to, or subtracted from, an even number of quadrants, the name of the function is the same in both columns; and when the arc is added to, or subtracted from, an odd number of quadrants, the names of the functions in the two columns are contrary in all cases, the algebraic sign is determined by the rules already given (Art. 58).

By means of this table, we may find the functions of any arc in terms of the functions of an arc less than 90° Thus,