Plane trigonometry, by S.L. Loney.

RATIOS OF A IN TERMS OF SIN A. 119
2
A
In a similar manner it may be shewn that when cos
is found from sin A, we should expect 4 values.
116. In any general case we can shew how the
ambiguities in relations (3) and (4) of Art. 113 may be
found.
We have
i. A, A= / ( 1 A 1 A\
sin + 2cos V2 2 + 2 - cos. A 7 A. 7-.A\
V2 Lsin cos - + cos - sin = 2 sin ( + -.
The right-hand member of this equation is positive if
7r A
+ - lie between 2n7r and 2n7r + tr,
4 2
A r 3w
i.e. if A lie between 2n7r - 7 and 2n7r + --.
2 4 4
A A A
Hence sin - + cos 2 is positive if - lie between
2 2 2
Vr 3vr
2n7r- - and 2n7r +;
4 4
it is negative otherwise.
Similarly we can prove that.A A. A 7r\
sin - -cos - = 2 sin 2-.
A A
Therefore sin A- cos - is positive if
2 i
iA 5r
i.e. if lie between 2n7r + 7 and 2n7r + -
2 4 4