1 9.2 - 9.3 the law of sines and the law of cosines in this chapter, we will work with oblique...
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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
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:
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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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.
Solving Case 2: SSA
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Two Triangles If and , two distinct triangles can be formed from the given information.
sinh b A a a b
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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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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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B
A
C
c
ba
2 2 2
2 2 2
2 2 2
2 cos
2 cos
2 cos
a b c bc A
b a c ac B
c a b ab C
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:
2 2 2 2 cosc a b ab C
2 2 22 cosab C a b c 2 2 2
cos2
a b cC
ab
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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Example: Solve ABC where a = 184, b = 125, and C = 27.2.
Solving Case 3: SAS
Give length to one decimal place and angles to nearest tenth of a degree.
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Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.
Solving Case 3: SAS
Give length to one decimal place and angles to nearest tenth of a degree.
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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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Example: Solve ABC where a = 128, b = 146, and c = 222.
Solving Case 4: SSS
Give angles to nearest tenth of a degree.
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When to use what……(Let bold red represent the given info)
Use Law of Sines
SSS
SSA
SAS
ASA
AAS
Use Law of Cosines
Be careful!! May have 0, 1, or 2 solutions.