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

PM sin a, OP' OP cos a, AT"" = AT=tan a,

=

BT" = BT = cot a, OT""=0T sec a, OT'—OT= =cosec a,

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

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

in question.

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°

PARTICULAR VALUES OF CERTAIN FUNCTIONS.

64. Let MAM' be any arc, denoted by 2a, M'M its chord, and OA a radius drawn perpendicular to M'M: then will PM = PM', and AM = AM' (B. III., P. VI.). But PM is the sine of AM, or, PM sin a: hence,

sin a = {M'M ;

M

M'

that is, the sine of an arc is equal to one half the chord

of twice the arc.

Let M'AM = 60°;

will equal the radius, or

then will AM = 30°, and M'M 1: hence, we have,

sin 30° =

that is, the sine of 30° is equal to half the radius.

FORMULAS EXPRESSING RELATIONS BETWEEN THE CIRCULAR FUNCTIONS OF DIFFERENT ARCS.

65. Let MB and BA represent two arcs, having the common radius 1; denote the first by u, and the second by b: then, MA=a+b. From M draw MP perpendicular to CA, and MN perpendicular to CB; from N draw NP' perpendicular to CA, and NL parallel to AC.

Then, by definition, we shall have,

PP'

PM sin (a + b), NM = sin a, and CN = cos α.

Since the triangle MLN is similar to CP'N (B. IV, P. 21), the angle LMN is equal to the angle P'CN; hence, from the right-angled triangle MLN, we have,

MLMN cos b = sin a cos b.

From the right-angled triangle CP'N (Art. 37), we have,

NP' CN sin b;

or, since NP' = LP, LP cos a sin b.

Substituting the values of PM, ML, and LP, in Equation (1), we have,

the

sin (a + b) = sin a cos b + cos a sin b; (A.). that is, the sine of the sum of two arcs, is equal to sine of the first into the cosine of the second, plus the sine of the first into the sine of the second.

co

Since the above formula is true for any values of a and -b, for b; whence,

b, we may substitute

sin (a - b) = sin a cos (b) + cos a sin ( − b) ;

that is, the sine of the difference of two arcs, is equal to the sine of the first into the cosine of the second, minus the cosine of the first into the sine of the second.

If, in Formula (B), we substitute (90° — a), for a, we have,

sin (90°-a-b) = sin (90°-a) cos b-cos (90° — a) sin b;

but (Art. 63),

hence, by substitution in Equation (2), we have,

cos (a + b) = cos a cos b - sin a sin b; (0.)

hat is, the cosine of the sum of two arcs, is equal to the ectangle of their cosines, minus the rectangle of their sines.

If, in Formula (), we substitute - b, for b, we find

cos (a - b) = cos a cos ( — b) sin a sin (b),

-