By a simple inspection of the figure, observing the rule 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:

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,

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

M

M'

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

Let M'AM = 60°; then will AM 30°, and M'M will equal the radius, or 1 hence, we have,

sin 30° = ;

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

Again, let M'AM = 90° : then will AM =
M'M = √2 (B. V., P. III.): hence, we have,

FORMULAS EXPRESSING RELATIONS BETWEEN THE CIRCULAR FUNCTIONS OF DIFFERENT ARCS.

65. Let AB and BM represent two arcs, having the common radius 1; denote the first by b, and the second by a. 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,

NB

PP'A

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

P'N CN sin b;

or, since

P'N PL,

=

PL = cos a sin b.

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

LM MN cos b = sin a cos b;

Substituting the values of PM, PL, and LM, in Equa tion (1), we have,

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 the sine of the first into the cosine of the second, plus the cosine of the first into the sine of the second.

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

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),

(2.)

sin (90°-a-b) = sin [90°— (a + b)] = cos (a + b),

and,

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

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

that is, the cosine of the sum of two arcs, is equal to the rectangle 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;