Plane trigonometry, by S.L. Loney.

204

TRIGONOMETRY.

From the figure of Art. 184 we have
b2 = c2 + a2 - 2ca cos B..'. a- 2ac cos B + c2 cos2 B = b2- c2 + c2 cos2 B
= b2 - c2 sin2 B..'. a - c cos B = + 2/b - c2 sin2 B,
i.e. a = c cos B + /b - c2 sin2 B............(1).
Now (1) is an equation to determine the value of a
when b, c and B are given.
(a) If b < c sin B, the quantity /b2 - c2 sin B is
imaginary and (1) gives no real value for a.
(/8) If b = c sinB, there is only one value, ccosB,
for a; there is thus only one triangle which is rightangled.
() If b > c sinB, there are two values for a. But,
since a must be positive, the value obtained by taking
the lower sign affixed to the radical is inadmissible unless
c cos B - /b2 - c2 sin2 B is positive,
i.e. unless /2 - c2 sin2 B < c cos B,
i.e. unless b2 - c2 sin2 B < c2 cos2 B,
i.e. unless b2 < c2.
There are therefore two triangles only when b is
> c sinB and at the same time < c.
188. Ex. Given b= 16, c = 25, and B = 33~ 15', prove that the triangle
is ambiguous and find the other angles, having given
log 2 = 30103, L sin 33~ 15' = 97390129,
L sin 58~ 56' = 9-9327616,
and L sin 58~ 57' = 9-9328376.