Download Law of Sines (LOS) Notes

Lesson 4.7
The Law of Sines
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
What you’ll learn about




Deriving the Law of Sines
Solving Triangles (AAS, ASA)
The Ambiguous Case (SSA)
Applications
… and why
The Law of Sines is a powerful extension of the triangle
congruence theorems of Euclidean geometry.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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Solving Oblique Triangles
In this section, we will work with oblique triangles
⇒ triangles that do NOT contain a right angle.
An oblique triangle has either:


three acute angles
or
two acute angles and one obtuse angle
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Every triangle has 3 sides and 3 angles.

To solve a triangle means to find the lengths of its
sides and the measures of its angles.

To do this, we need to be given at least three of these
parts, and at least one of them must be a side.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Here are the four possible combinations of parts:
1.
Two angles and one side (ASA or SAA)
2.
Two sides and the angle opposite one of them (SSA)
3.
Two sides and the included angle (SAS)
4.
Three sides (SSS)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Case 1:
Two angles and one side (ASA or SAA)
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Case 2:
Two sides and the angle opposite one of
them (SSA)
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Case 3:
Two sides and the included angle (SAS)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Case 4:
Three sides (SSS)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The Law of Sines
a
b
c
= =
sin A sin B sin C
In  ABC with angles A, B, and C opposite sides a, b, and c, respectively,
the following equation is true:
sin A sin B sin C
= =
.
a
b
c
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Case 1: ASA or SAA
Give lengths to two decimal places.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Case 1: ASA or SAA
Give lengths to two decimal places.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Case 2: SSA

In this case, we are given two sides and an angle opposite.

This is called the AMBIGUOUS CASE.
That is because it may yield no solution, one solution, or
two solutions, depending on the given information.

Please refer to the chart in your Google Drive
Folder
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
SSA --- The Ambiguous Case
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
b sin A, then side a is not
If a < h =
sufficiently long enough to form a triangle.
No Triangle
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
If a= h= b sin A , then
side a is just long enough to form a right triangle.
One Right Triangle
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Two Triangles=
If h b sin A < a
and a < b, two
distinct triangles can be formed from the given
information.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
One Triangle
If a ≥ b , only one triangle can be
formed.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Give lengths to two decimal places and angles to nearest tenth of a degree.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Give lengths to two decimal places and angles to nearest tenth of a degree.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Give lengths to two decimal places and angles to nearest tenth of a degree.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example: Solve ∆ABC where A = 27.6°, a =112, and c = 165.
Give lengths to two decimal places and angles to nearest tenth of a degree.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
To deal with Case 3 (SAS) and Case 4 (SSS), we do not
have enough information to use the Law of Sines.
So, it is time to call in the Law of Cosines.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example Solving a Triangle Given Two Sides
and an Angle (The Ambiguous Case)
= 6, and ∠A
= 30 .
Solve  ABC given that a= 7, b

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 5- 24