Plane trigonometry, by S.L. Loney.

114

TRIGONOMETRY.

Now 165~ lies between 90~ and 180~, so that, by Art. 52, its sine is
positive and its cosine is negative.
Hence sin 165~ =/3 -1
and cos 165~= - ^./3 -From the above examples it will be seen that, when the angle A and
its cosine are given, the ratios for the angle A may be determined without
any ambiguity of sign.
When however only cos A is given, there is an ambiguity in finding
A A
sin2 and cos. The explanation of this ambiguity is given in the next
article.
##112. To explain why there is ambiguity when cos A
A
and sin - are found from the value of cos A.
We know that, if n be any integer,
cos A = cos (2n7r + A) = k (say).
A
Hence any formula which gives us cos 2 in terms of k,
2n'zr + A
should give us also the cosine of 2n
2
2n7r +A { A\
Now cos 2_- =Cs 2+)
A.. A A
= cos n7r cos - + sin nr sin - = cos nr cos
2 2 2
A
= cos -
according as n is even or odd.