Transcript Section 5-5

Section 5-5
Law of Sines
Section 5-5
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solving non-right triangles
Law of Sines
solving triangles AAS or ASA
solving triangles SSA
Applications
Solving Non-Right Triangles
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In Chapter 4 we learned to solve right
triangles, which meant that we could find all
its missing parts
In the next two sections, we will learn to solve
non-right triangles using two new tools: the
Law of Sines and the Law of Cosines
Which tool we use depends upon which parts
we are given
Solving Non-Right Triangles
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Law of Sines
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ASA – two angles and the included side
AAS – two angles and a non-included side
SSA – two sides and a non-included angle
Law of Cosines
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SSS – three sides
SAS – two sides and the included angle
Law of Sines
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For any ΔABC with angles A, B, and C and
opposite sides a, b, and c, respectively:
sin A sin B sin C
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a
b
c
or
a
b
c
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sin A sin B sin C
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Either version can be used, but it is easier if
the missing variable is in the numerator
AAS and ASA
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since both AAS and ASA form a triangle
congruence (from geometry), there is exactly
one triangle that can be formed using the
three parts
when solving either of these types of
triangles, you are looking for only one
possible solution
Solve ABC given A  32, B  45, and a  9.
SSA – The Ambiguous Case
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you might recall from geometry that SSA is
not a congruence postulate or theorem
If given SSA, there are three possible results
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No triangle is formed
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One triangle is formed
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Two triangles are formed
Solve ABC given A  25, b  6, and a  7.
Solve ABC given A  30, b  7, and a  6.
A forest ranger at ranger station A sights a fire in the
direction 32 east of north. A ranger at station B, 10 miles
due east of A, sights the same fire on a line 48 west of
north. Find the distance from each ranger station to the fire.