on a straight line making an angle + -}-ir with OD, and is therefore
equal to OF sin B; hence
OF cos (A -l-B) =-'OE cos A-I-EF cos (%1r+A) =OF (cos A cos B-sin A. sin B),
or cos (A +B) ==cos A cos B-sin A sin B. The angles A, B are absolutely unrestricted in magnitude, and thus this formula is perfectly general. We may change the sign of B, thus cos (A -B)=cos A cos (~B)-sin A sin (-B), or cos (A -B) =cos A cos B»|-sin A sin B. If we projected the sides of the triangle OEF on a straight line making an angle-I-if with OA we should obtain the formulae sin (A iB) =sin A cos Bicos A sin B, which are really contained in the cosine formula, since we may put %'ll""'B for B. The formulae
tanA=f=tanB cotAeotB=»=I
im (A *B> immma '=°' <A *Bl =@a§ 7r are immediately deducible from the above formulae. The equations HC-it-D) cos %(C-D),
§ (C-D) cos'§ (C-l-D),
sin C-{-sin D=2 sin
sin C-sin D=2 sin
cos D-l-cos C=2 cos § (C-l-D) cos é(C-D), cos D-cos C=2 sin %(C-l-D) sin § (C-D), may be obtained directly by the method of projections. Take two equal straight lines OC, OD, making angles C, D, with OA, and draw OE perpendicular to CD. The angle which OE makes c with OA is HC-I-D) and that which E DC makes is'1r C D'the anle
if -l' 'l' ly S
COE is %(C~D). The sum of the pro-
jections of OD and DE on OA is equal to that of OE, and the sum of the
projections of OC and CE is equal to that of OE; hence the sum of the projections of OC and OD is twice that of
OE, or cos C4-cos D=2 cos § (C+D)
cos %(C-D). The difference of the
0 A projections of OD and OC an OA
is equal to twice that of ED, hence we have the formula cos D-cos C
D The other two formulae will
FIG. 4.
=2 sin %(C-l-D) sin %(C-).
be obtained by projecting on a straight line inclined at an angle j“%7l' to OA.
As another example of the use of projections, we will find the sum of the series cos a.-I-cos (a-|-H)-l-cos (a+2;3)+, . ~}-cos (a-l-n-IB). S of § uppose an unclosed polygon each angle of which 52:6 of is -r-B to be inscribed in a circle, and let A, A, , A2, cosines In A3, . . ., A., be n~}-I consecutive angular points; Amhmeucal let be the diameter of the circle; and suppose a progression straight line drawn making an angle a. w1th AA1, then a-l-B, a-l"2B, . are the angles it makes with A1 A3, /12, A3, ...; wtiave by projections AA" cos (a-I-inf 1B)=AA1{cos a+ cos a+B+...+cosa+(n-1)B}, also ' AA1=D sin id, AA, ,=D sin %7lB; hence the sum of the series of cosines is cos (a.~l-én-I 13) sin $1113 cosec é/3. By a double application of the addition formulae we may obtain the formulae
sin (A, -i-A24-A3) =sin A1 cos A2 cos A3 F°""""" f°' ~l-cos A1 sin A3 cos A34-cos A, cos A3 sin A3 Slneand -A -A -A
Cosmeof -sm, sin gsm 3,
Sum of cos (Al-l-A3-l-A3) =cos A3 cos A3 cos A3 Angles -cos A1 sin A3 sin A3-sin A1 cos A2 sin A3 -sin A1 sin A3 cos A3.
We can by induction extend these formulae to the case of n angles. Assume sin (A1+A2+ . . +A, ,)=S1-S3+Sf, -, . . cos (A1+A3+ . . +A, ,)=S3-S2+S4- . . . where S, denotes the sum of the products of the sines of r of the angles and the cosines of the remaining 11-1 angles; then we have Sill . . . *l'An'l'A, ,+1)=Cf)S An+1(S1"S3"j'S5" . . -l-S111 An+1(S0-'S2“l"S4'- .
The rght-hand side of this equation may be written (S, cos A, .,1+.S'., sin A, ,+1) - (S3 cos A, ,+1-{-S3 sin A, .+1)+ . ., Of SQ-S'3+ .
where SC denotes the quantity which corresponds for n-H angles to Sf for n angles; similarly we may proceed with the cosine formula. The theorems are true for n=2 and n=3; thus they are true generally. The formulae
F°"'""'<"° cos 2A =cos” A -sin2A =2 cos' A - I = I -2 sin” A, for Multiple
t A
Jndsub' sin 2A =2 sin A cos A, tan 2A =»-1-%-, Mumple ' *tan
Angles. sm 3 4 =3 sin A -4 sin* A, cos 3A =4 cos3 A -3 cos A, sin nA = n cos"'1 A sin A - cos”'° A sin' A + . . +<- nf"—-l" ' bl, jg, ff' "”) cw-'=f-1 A Sin2f+1 A, n- 1 .
cos nA =cos"A -MET) cos"" A sin' A + . . +()f) Cosa-ar A Sinn A+ 5
may all be deduced from the addition formulae by making the angles all equal. From the last two formulae we obtain by division A ntanA - ¢an2A+...+<-1>f ¢an2f+1A+.., tan n f ' ' -
1 -7%-f-Qtan2A-l-...+(~1>f >¢an2fA+... 3 tan A -tan" A
1 -3 tan2 A
The values of sin é/1, cos § A, tan %A are given in terms of cos A by the formulae
-, A t 1 A 3
sm;/1=<-1>@ 1-¥-, ¢0s», eA=<-na iE'f-, In the particular case of n=3 we have tan 3A = - A
- ani A =<“>' ¥;§ 7 »
where P is the integral part of A/2-/r, q the integral part of A/21r--ir, and r the integral part of A /1r.
Sin %A, cos 5A are given“in terms of sin A by the formulae 2 sin 35A = (-1)l>'(I -l-sin A)%+(-I)q'(I —sin A)l, 2 cos %A = (-1)i>'(1--sin AH-(-1)'1'(I -sin A)l, Xfliere ji' is the integral part of A /27l"'j'% and q' the integral part of zur-~.
6. Iniany plane triangle ABC we will denote the lengths of the sides BC, CA, AB by a, b, c respectively, and the angles BAC, ABC, A CB by A, B, C respectively. The fact that the projections of b and 1: on a straight line perpendicular to the P"°p"“" side a are equal to one another is expressed by the equa- of Triangles tion b sin C=c sin B; this equation and the one obtained by projecting c and a on a straight line perpendicular to a may be written a/sin A =b/sin B =c/sin C. The equation a=b cos C-l-c cos B expresses the fact that the side a is eqlual to the sum of the projections of the sides b and 6 on itself; t us we obtain the equations
a=bcos C+c cos B*
b=ccosA-1-acos Cc=acosB-|-bcosA
If we multiply the first of these equations by -a, the second by b, and the third by c, and add the resulting equations, we obtain the formula 122-|-of-a2=2bc cos A or cos A =(b2-{-cl-112)/2bc, which gives the cosine of an angle in terms of the sides. From this expression for cos A the formulae
Sin %A= (S-bgés-c) E, COS %%1= 5(sZ;a) g é, . tan;/1 = Sggfgi i, Sin A =, %;§ <s-a><s-bm-¢>}*, where s denotes Ha-l-b-l-6), can be deduced by means of the dimidiary formula.
From any general relation between the sides and angles of a triangle other relations may be deduced by various methods of transformation, of which we give two examples. o.. In any general relation between the sines and cosines of the angles A, B, C of a triangle we may substitute pA +qB+1'C, rA +pB+qC, QA -l-VB-1-pC for A, B, C respectively, where p, q, 1' are any quantities such that p-l-q-I-r-l-I is a positive or negative multiple of 6, provided that we change the signs of all the sines. Suppose p-I-q-l-r-l-I =6n, then the sum of the three angles 2mr - (pA +qB+rC),2n1r - (rA +pB +qC),21z1r - (gA -I-1B-I-pC) is 1r; and, since the given relation follows from the condition A+B-l-C =-fr, we may substitute for A, B, C respectively any angles of which the sum is 1r; thus the transformation is admissible. B. It may easily be shown that the sides and angles of the triangle formed by joining the feet of the perpendiculars from the angular points A, B, C on the opposite sidjes of the triangle ABC are respectively a cos A, b cos B, c cos C,1r-2A,1r-2B, '/r-2C; we may therefore substitute these expressions for a, b, 6, A, B, C respectively in any general formula. By drawing the perpendiculars of this second triangle and joining their feet as before, we obtain a triangle of which the sides are-a cos A cos 2A, - b cos B cos 2B, -L cos C cos 2C and the angles are 4A -1r, 4B~jvr, 4C—/r; We may therefore substitute these expressions for the sides and angles of the original triangle; for example, we obtain thus the formula COS A a2 cos” A cos” 2A -122 cos' B cos? 2B-02 cosz C cos' 2C 4 2bc cos B cos C cos 2B cos 2C
This transformation obviously admits of further exten-SIOU- Solution of
(I) The three sides of a triangle ABC being given, Triangles. the angles can be determined by the formula L tan éA = I0-j-§ log (s-b)+§ log (s-c)-é log s-§ log (s»a) and two corresponding formulae for the other angles.
