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 a, and the second by b; then, AM = a + b. From M draw PM perpendicular to CA, and NM perpendicular to CB; from N draw NP' perpendicular, and NL parallel, to CA.

Then, by definition, we have

PM sin (a + b), NM = sin b,

From the figure, we have

PM PL + LM.

NB

PPA

and CN cos b.

(1.)

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

PL cos b sin a = sin a cos b.

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

LM NM cos a = sin b cos a = cos a sin b. Substituting the values of PM, PL, and LM, in equation (1), we have

(A.)

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

b

cos a sin b;

(B.)

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

that is, the sine of the difference of two ares 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; (2.)

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

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

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 (C), member by member, 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 (E), 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 (C) by formula (A), member by member, and reducing, we have

and thence, by the substitution of b for b,

b =

66. If, in formulas (A), (C), (E), and (G), we make =a, we find

71

Dividing equation (A'), first by equation (4), and then by equation (3), member by member, we have

Substituting a for a, in equations (1), (2), (5), and

(6), we have

Taking the reciprocals of both members of the last two

formulas, we have also,

67. If formulas (A) and (B) are first added, member to member, and then subtracted, member from member, and the same operations are performed upon (C) and (D),

we obtain