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 Thus,

arc less than 90°

PARTICULAR VALUES OF CERTAIN FUNCTIONS.

61. 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

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

M'M = √2 (B. V., P. III.): hence, we have,

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 , and the second by b: then, MA=a+b. From M draw MP perpendicular to CA, and MV perpendicular to CB; from N draw NP' perpendicular to CA, and NL parallel to AC.

Then, by definition, we shall have,

C

PP'A

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

From the figure, we have,

PMML + LP.

(1).

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,

ML = MN 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 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° ab) = sin [90° — (a + b)] = cos (a + b),

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 sincs.

that is, the cosine of the difference of two arcs, is equal to the rectangle of their cosines, plus the rectangle of their sines.

If we divide Formula (A) by Formula (@), member by nember, we have,

Dividing both terms of the second member by cos a cos b, recollecting that the sine divided by the cosine is equal to the tangent, we find,

that is, the tangent of the sum of two arcs, is equal to the sum of their tangents, divided by 1 minus the rectangle of their tangents

If, in Formula (), we substitute b, for b, recollecting that

tan (-b)

=

tan b, we have,

that is, the tangent of the difference of two arcs, is equal to the difference of their tangents, divided by 1 plus the rectangle of their tangents.

In like manner, dividing Formula () by Formula (A), member by member, and reducing, we have,