Plane trigonometry, by S.L. Loney.

RATIOS OF - IN TERMS OF SIN A.
2

115

A
Similarly any formula giving us sin A in terms of k,
2
2nw' + A
should give us also the sine of 2n.
Also sm 2- = sin r + -
Ao sin 2n.'+ A =sin (- A~
2 2
= sin nr cos - + os n7r sin = + cos nwr sin
n 2 2
= +sin
Hence in each case we should expect to obtain two
A A
values for cos and sin -, and this is the number which
2 2
the formulae of Art. 110 give.
113. To express the trigonometrical ratios of the angle
A.
- in terms of sin A.
From equation (1) of Art. 109 we have
sinA A(1).
2 sin 2 cos - = sin A............... (1).
A A
Also sin2 + cos2 -= 1, always.......... (2).
2 2 ways. (2
First adding these equations, and then subtracting
them, we have.A A A A
sin2 + 2 sin - cos - + cos2 = 1 + sin A,
2 2 2 2
A A A
and Sin2 A 2 sin- + cos2 = 1 - sin A;
~~~2 2 2 2
8-2