Plane trigonometry, by S.L. Loney.

AMBIGUOUS CASE.

203

touch BD (Fig. 2), or it may meet BD in two points CU
and C., (Figs. 3 and 4).
In the case of Fig. 1, it is clear that there is no
triangle satisfying the given condition.
Here b < AD, i.e. < c sin B.
In the case of Fig. 2, there is one triangle A BD
which is right-angled at D. Here
b = AD= c sin B.
In the case of Fig. 3, there are two triangles ABC,
and ABC,. Here b lies in magnitude between AD and c,
i.e. b is > c sin B and < c.
In the case of Fig. 4, there is only one triangle ABC,
satisfying the given conditions [the triangle ABC2 is
inadmissible; for its angle at B is not equal to B but is
equal to 180~- B]. Here b is greater than both csin B
and c.
To sum up:
Given the elements b, c, and B of a triangle,
(a) If b be < c sin B, there is no triangle.
(/) If b = c sin B, there is one triangle right-angled.
(7y) If b be > c sin B and < c and B be acute, there are
two triangles satisfying the given conditions.
(8) If b be > c, there is only one triangle.
Clearly if b = c, the points B and C2 in Fig. 3 coincide
and there is only one triangle.
If B be obtuse, there is no triangle except when b > c.
187. The ambiguous case may also be considered
algebraically as follows.