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The Law of Tangents
•
Section 2.3
53
Example 2.12
Can a triangle have the parts
a
=
6,
b
=
7,
c
=
9,
A
=
55
◦
,
B
=
60
◦
, and
C
=
65
◦
?
Solution:
Before using Mollweide’s equations, simpler checks are that the angles add up to 180
◦
and
that the smallest and largest sides are opposite the smallest and largest angles, respectively. In this
case all those conditions hold. So check with Mollweide’s equation (2.22):
a
+
b
c
=
cos
1
2
(
A
−
B
)
sin
1
2
C
6
+
7
9
=
cos
1
2
(55
◦
−
60
◦
)
sin
1
2
(65
◦
)
13
9
=
cos (
−
2.5
◦
)
sin 32.5
◦
1.44
=
1.86
✗
Here the difference is far too large, so we conclude that there is no triangle with these parts.
We will prove the Law of Tangents and Mollweide’s equations in Chapter 3, where we will
be able to supply brief analytic proofs.
6
Exercises
For Exercises 13, use the Law of Tangents to solve the triangle
△
ABC
.
1.
a
=
12,
b
=
8,
C
=
60
◦
2.
A
=
30
◦
,
b
=
4,
c
=
6
3.
a
=
7,
B
=
60
◦
,
c
=
9
For Exercises 46, check if it is possible for a triangle to have the given parts.
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.
 Fall '11
 Dr.Cheun
 Calculus, Equations, Angles, Geometric Proofs

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