Plane trigonometry, by S.L. Loney.

118

TRIGONOMETRY.

A
Hence any formula which gives us sin - in terms of k,
should give us also the sine of n + ( -1) A
2
First, let n be even and equal to 2m. Then
si. 7r+(-l)nA in. / A
sm 2
A.A A
= sin m7r cos - + cos mr sin = cos mrr sin
= sin
2'
according as m is even or odd.
Secondly, let n be odd and equal to 2p + 1.
Then
i nr + (_ l)n A sin 2p7r + rr-A = in r-r - A
2 2 2=
r'-A. rr-A A
= sinpr cos - + cosp sin - =cos p7 cos
2'
A
= cos -,
according as p is even or odd.
Hence any formula which gives us sin A in terms of
sin A should be expected to give us, in addition, the
values of
A A A
-sin-, cos- and -cos
i.e. 4 values in all. This is the number of values which
we get from the formulae of Art. 113, by giving all possible
values to the ambiguities.