9.2 - 9.3 The Law of Sines and The Law of Cosines

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9.2 - 9.3 The Law of Sines and The Law of
Cosines

• In these sections, we will study the following
  topics
• Solving oblique triangles (4 cases)
• Solving a triangle using Law of Sines
• Solving a triangle using Law of Cosines

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• 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
• two acute angles and one obtuse angle

or
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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
• Two angles and one side (ASA or SAA)
• Two sides and the angle opposite one of them
  (SSA)
• Two sides and the included angle (SAS)
• 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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The Law of Sines
Three equations for the price of one!
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Solving Case 1 ASA or SAA
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Solving Case 1 ASA or SAA
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Example using Law of Sines

• A ship takes a sighting on two buoys. At a
  certain instant, the bearing of buoy A is N
  44.23 W, and that of buoy B is N 62.17 E. The
  distance between the buoys is 3.60 km, and the
  bearing of B from A is N 87.87 E. Find the
  distance of the ship from each buoy.

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Continued from above
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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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No Triangle If , then
side is not sufficiently long enough to form a
triangle.
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One Right Triangle If ,
thenside is just long enough to form a right
triangle.
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Two Triangles If and
, two distinct triangles can be formed
from the given information.
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One Triangle If , only one triangle can
be formed.
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Continued from above
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Continued from above
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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.
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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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The Law of Cosines
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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
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Guidelines for Solving Case 3 SAS

• When given two sides and the included angle,
  follow these steps
• Use the Law of Cosines to find the third side.
• 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.)
• 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?.
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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?.
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Continued from above
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Guidelines for Solving Case 4 SSS

• When given three sides, follow these steps
• Use the Law of Cosines to find the LARGEST ANGLE
  (opposite the largest side).
• Use the Law of Sines to find either of the two
  remaining angles.
• 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.
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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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Continued from above
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• End of Sections 9.2 9.3