Plane trigonometry, by S.L. Loney.

134

TRIGONOMETRY.

Ex. Write down the value of tan 40.
Here tan 40 S-S 3 4taln-403tan3
1-s2+4 1 - 4C tan2 0+4C4 tan4 0
4 tan 0 - 4 tan3 0
1 - 6 tan'0 + tan4 0
x. Prove that tan 5 tan 0 - 10 tan3 0 + tan5 0
1 - 10 tan2 0 + 5 tan4 0
126. By a method similar to that of the last article
it may be shewn that sin (A, + A2 +... + An)
= cos AI cos A2... cos An (si- S3 + -...),
and that cos (A A+ A... + A)
= cos Al cos A2... cos An (1 - s2 + s4-...),
where si, s,, s3,... have the same values as in that article.
127. Identities holding between the trigonometrical ratios of the angles of a triangle.
When three angles A, B and C, are such that their
sum is 180~, many identical relations are found to hold
between their trigonometrical ratios.
The method of proof is best seen from the following
examples.
Ex. 1. If A + B +0 = 180~, to prove that
sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.
sin 2A + sin 2B + sin 2C
=2 sin (A +B) cos (A - B) + 2 sin C cos C.
Since A + B + C= 180,
we have A + B=180~ - C,
and therefore sin (A + B) = sin C,

and

cos (A + B)= - cos C.

(Art. 72)