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52
Chapter 2
•
General Triangles
§2.3
Note that in any triangle
△
ABC
, if
a
=
b
then
A
=
B
(why?), and so both sides of formula
(2.17) would be 0 (since tan 0
◦
=
0). This means that
the Law of Tangents is of no help in
Case 3 when the two known sides are equal
. For this reason, and perhaps also because of the
somewhat unusual way in which it is used, the Law of Tangents seems to have fallen out of
favor in trigonometry books lately. It does not seem to have any advantages over the Law
of Cosines, which works even when the sides are equal, requires slightly fewer steps, and is
perhaps more straightforward.
4
Related to the Law of Tangents are
Mollweide’s equations
:
5
Mollweide’s equations
: For any triangle
△
ABC
,
a
−
b
c
=
sin
1
2
(
A
−
B
)
cos
1
2
C
, and
(2.21)
a
+
b
c
=
cos
1
2
(
A
−
B
)
sin
1
2
C
.
(2.22)
Note that all six parts of a triangle appear in both of Mollweide’s equations. For this
reason, either equation can be used to check a solution of a triangle. If both sides of the
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 Fall '11
 Dr.Cheun
 Calculus, Angles

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