College Algebra & Trigonometry, 2018a

478 CHAPTER 11. THE LAW OF SINES THE LAW OF COSINES
11.2 The Law of Sines: the ambiguous case
In all of the examples and problems in Section 4.1, notice that we were always
given two angles and one side, although we could use the Law of Sines if we
were given one angle and two sides (as long as one of the sides corresponded to
the given angle). This is because when we use the Law of Sines to find an angle,
an ambiguity can arise due to the sine function being positive in Quadrant I and
Quadrant II.
We saw in Chapter 3 that multiple answers arise when we use the inverse trigonometric
functions. For problems in which we use the Law of Sines given one angle
and two sides, there may be one possible triangle, two possible triangles or no
possible triangles. There are six different scenarios related to the ambiguous case
of the Law of Sines: three result in one triangle, one results in two triangles and
two result in no triangle.
One Triangle
a>b
a = h
b
h
a
b
h
a
A
A
a>b
b
a
A