Plane trigonometry, by S.L. Loney.

326

TRIGONOMETRY.

Hence equating real and imaginary parts, we have
sin(a +/ +...)=cos a cos/3cos y... [s - s3+ 85- s7...]...(3),
and
cos(a+,8 +7y...)=cosacos cos y...(1 -s + s4-s...)...(4).
Hence, by division,
tan (a+ + +..) y + s s..) S..(5).
1 — S2 + S4 - S6..
The signs in the expressions on the right hand of (3)
and (4) are alternately positive and negative.
The relation (5) was shewn, by Induction, to be true
in Art. 125.
278. Ex. Prove that the equation
a2 cos2 + b2sin2 + 2gacos + 2fb sin + c = O
has 4 roots, and that the sum of the values of 0 which satisfy it is an even
multiple of ir radians.
Let t - tan.
6 6
2 tan 1- tan2
Then since (Art. 109), sin 0= and cos 0 =
1 +tan2 1 + tan2
2 2
the equation above becomes
a2 1 t2) b2 2t 2) 1 t2 f2
a 13-t" ibz(^fi + 2ga + 2fb' -- + =O
or, on reduction and simplification,
t4 (a2- 2ga c) + 4fbt3+ t2(4b - 2a2 2c) + 4fbt + a2 2ga c=0......(1).
This is an equation having 4 roots.
4fb
Also s, =sum of the roots= - -4
a2 - 2ga + c '
4b2 - 2a2 + 2c
s2 = sum taken two at a time = a 2ga + c
a2 -2ga+c '
4fb,
s3 =sum taken three at a time = - 42 - b
and S4= sum taken four at a time=a2-2ga+c
a - 2ga + c