Plane trigonometry, by S.L. Loney.

AMBIGUOUS CASE.

201

185. The equation (1) of the previous article gives in
some cases no value, in some cases one, and sometimes
two values, for C.
If c sin B > b, the right-hand member of (1) is greater
than unity, and hence there is no corresponding value
for C.
If c sinB =b, the right-hand member of (1) is equal
to unity and the corresponding value of C is 90~.
c sin B
If c sin B < b, there are two values of C having c, —
as its sine, one value lying between 0~ and 90~ and the
other between 90~ and 180~.
Both of these values are not however always admissible.
For if b > c, then B > C. The obtuse-angled value of
C is now not admissible; for, in this case, C cannot be
obtuse unless B be obtuse also, and it is manifestly
impossible to have two obtuse angles in a triangle.
If b < c and B be an acute angle, both values of C
are admissible. Hence there are two values found for
A and hence the relation (2) gives two values for a. In
this case there are therefore two triangles satisfying the
given conditions.
Since, for some values of b, c and B, there is a doubt
or ambiguity in the determination of the triangle, this
case is called the Ambiguous Case of the solution of
triangles.
186. The Ambiguous Case may also be discussed
in a geometrical manner.
Suppose we were given the elements b, c and B and
that we proceeded to construct, or attempted to construct,
the triangle.