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9.2 - 9.3 The Law of Sines and The Law of Cosines
In this chapter, 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
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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 know at least three of these
parts, and at least one of them must be a side.
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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)
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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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Case 3:
Two sides and the included angle (SAS)
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Case 4:
Three sides (SSS)
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C
b
a
B
A
c
The Law of Sines
a
b
c


sin A sin B sin C
Three equations for the price of one!
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Solving Case 1: ASA or SAA
Give lengths to two decimal places.
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Solving Case 1: ASA or SAA
Give lengths to two decimal places.
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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.
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SSA --- The Ambiguous Case
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If a  h  b sin A, then side a is not
sufficiently long enough to form a triangle.
No Triangle
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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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If h  b sin A  a and a  b, two
distinct triangles can be formed from the given
information.
Two Triangles
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One Triangle
If a  b, only one triangle can be
formed.
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Give lengths to two decimal places and angles to nearest tenth of a degree.
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Continued from above
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Give lengths to two decimal places and angles to nearest tenth of a degree.
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Continued from above
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Give lengths to two decimal places and angles to nearest tenth of a degree.
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Making fairly accurate sketches can help you to determine the
number of solutions.
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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.
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Continued from above
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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.
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C
b
a
B
A
The Law of Cosines
c
a  b  c  2bc cos A
2
2
2
b  a  c  2ac cos B
2
2
2
c  a  b  2ab cos C
2
2
2
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Using Law of cosines to Find the Measure of an Angle
*To find the angle using Law of Cosines, you will need to solve
the Law of Cosines formula for CosA, CosB, or CosC.
For example, if you want to find the measure of angle C, you
would solve the following equation for CosC:
c  a  b  2ab cos C
2
2
2
2ab cos C  a  b  c
2
2
a b c
cos C 
2ab
2
2
2
2
To solve for angle C, you would
take the cos-1 of both sides.
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Guidelines for Solving Case 3: SAS
When given two sides and the included angle, follow
these steps:
1.
Use the Law of Cosines to find the third side.
2.
Use the Law of Cosines to find one of the
remaining angles.
You could use the Law of Sines here, but you must be careful due to
the ambiguous situation. To keep out of trouble, find the SMALLER of
the two remaining angles (It is the one opposite the shorter side.)
3.
Find the third angle by subtracting the two
known angles from 180.
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Solving Case 3: SAS
Example: Solve ABC where a = 184, b = 125, and C = 27.2.
Give length to one decimal place and angles to nearest tenth of a degree.
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Continued from above
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Solving Case 3: SAS
Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.
Give length to one decimal place and angles to nearest tenth of a degree.
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Continued from above
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Guidelines for Solving Case 4: SSS
When given three sides, follow these steps:
1.
Use the Law of Cosines to find the LARGEST
ANGLE (opposite the largest side).
2.
Use the Law of Sines to find either of the two
remaining angles.
3.
Find the third angle by subtracting the two
known angles from 180.
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Solving Case 4: SSS
Example: Solve ABC where a = 128, b = 146, and c = 222.
Give angles to nearest tenth of a degree.
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Continued from above
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When to use what……
(Let bold red represent the given info)
SAS
AAS
ASA
Be careful!!
May have 0, 1, or
2 solutions.
SSS
SSA
Use Law of Sines
Use Law of Cosines
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Give lengths to two decimal places and
angles to nearest tenth of a degree.
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Continued from above
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End of Sections 9.2 – 9.3
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