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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:

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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BA

C

c

ba

sin sin sin

a b c

A B C

Three equations for the price of one!

The Law of Sines

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Solving Case 1: ASA or SAA

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Solving Case 1: ASA or SAA

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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.

Example using Law of Sines

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Continued from above

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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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SSA --- The Ambiguous Case

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No Triangle If , then side is not sufficiently long enough to form a triangle.

sina h b A a

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One Right Triangle If , thenside is just long enough to form a right triangle.

sina h b A a

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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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One Triangle If , only one triangle can be formed.

a b

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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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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 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

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Continued from above

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Example: Solve ABC where b = 16.4, c = 10.6, and A = 128.5.

Solving Case 3: SAS

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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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Example: Solve ABC where a = 128, b = 146, and c = 222.

Solving Case 4: SSS

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Continued from above

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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.

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Continued from above

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End of Sections 9.2 – 9.3