(13.)

Formula relating to three arcs.

sin (a+b+c)=sin a.cos b. cos c + cos a. sin b. cos c

+ cos a.cos b. sin c― sin a.sin b. sin c.

tan (a+b+c)

(14.) Relations triangles.

tan atan b+ tan c-tan a.tan b. tan c 1- (tan a. tan b+ tan a'tan c + tan b. tan c)

between the sides and angles of plane

sin 24+ sin 2 B + sin 2 C = sin A. sin B. sin C.

cos 2 4 + cos 2 B + cos 2 C4 cos A. cos B. cos C-1.

tan A+tan B+tan C=tan A. tan B. tan C.

1

2

Let R1 be the radius of the inscribed, and R2 that of the circumscribing circle; then

(15.)

R1 = } (s — a) (s — b) (s — c)]3 ;

= s.tan A.tan 1⁄2 B.tan 1 C.

R2=abc.s (sa) (s—b) (s—c) ||

2

(Legendre, Geom. Note v; Hind, Trig. 153-64.) Values of the side c.

(a-b)+4 ab (sin C)

b cos 4+ a. 1- (sin B)2

b cos 4 ± a2 − (b sin 4)o]*.

a

cos B + sin B.cot C

Values of sin C.

sin B

{a cos B + b2 — (a sin B)*}.

a (sin B)2 + cos B. bo — (a sin B)o |3

* Analogous formulæ may be obtained by substituting A, a, for B, b,

and vice versa, in this, and several of the following formula.

Right angled triangles: C, the right angle.

(19.)

log blog c + log Cos A — log r.

or b may be determined as above.

log clog r+log blog Cos A;

(W. Ch. v; L. 67-71; Leg. 48-52; C. 529-58.)

*The sine, cosine, &c. to the tabular radius r will, for the sake of distinction, be denoted by capital letters. See (Enc. Met. Art. Trigonom.)

(20.) Solution of oblique angled plane triangles.

log Sin 4= log a log b + log Sin B.

This result is ambiguous if a > b, and B is an acute angle.

log Tan 1⁄2 (A–B) = log (a−b) — log (a+b)+log Tan 1⁄2 (A+B), A+B=180°- C:

or thus: suppose a > b, and let tan 0:

log Tan 0=log r + log a— log b: then

=

a

tan (A-B)=tan (A+B). tan (0— 45o),

2

log Tan (A-B)=log Tan (A+B)+log Tan (0—45°) — logr. knowing A + B, and A – B, A and B are determined.

log Tan 0=1{log 2+log a+log b+log Vers C+log r}−log (a—b);

log Sin0=1{log2+loga+logb+log(r+Cos C)+logr}— log(a+b);

+ {logs + log (s-a) + log (s—b) + log (s—c)}.

(s—b) (s—c).

Second Method: (sin 4)2=

;

bc

log Sin 4=1{log (s—b) + log (s—c) - log blog c} + log r.

Third Method: (cos 14)°— § (s—a);

bc

log Cos A= {log s + log (s — a) — log b — log c} + log r.