Unit 2  Objective 5  Oblique Triangles & Laws of Sines
Quick Review
Triangles that do not have a right angle are called oblique triangles. The trigonometric methods for solving right triangles do not work with oblique triangles. There are two methods that are usually used with oblique triangles: law of sines and law of cosines (objective 6).
The law of sines
The law of sines are used to solve the following two cases:
 When the measure of two sides and the angle opposite one of them is known, and
 When the measure of two angles and one side is known.
Example 1
Given: A = 33°, a = 9.4 and c = 14.3
Solve the triangle.
Solution
We are given: 

sides  angles 
a = 9.4  A = 33° 
b = ?  B = ? 
c = 14.3  C = ? 

Using the law of sines:
B = 180°  33°  55.95° = 91.05°
We can now find the length of the third side.
Now we have: 

sides  angles 
a = 9.4  A = 33° 
b = 17.26  B = 91.05° 
c = 14.3  C = 55.96° 

Example 2
Given: A = 82.17°, B = 64.43° and c = 9.12
Solve the triangle.
Solution
We are given: 

sides  angles 
a = ?  A = 82.17° 
b = ?  B = 64.43° 
c = 9.12  C = ? 

We can find angle C:
C = 180°  82.17°  64.43° = 33.40°
Now use the law of sines:
Now find side b using the law of sines:
Now we have: 

sides  angles 
a = 16.41  A = 82.17° 
b = 14.94  B = 64.43° 
c = 9.12  C = 33.40° 

Example 3
Given: a = 20, b = 24, and A = 55.4°
Solve the triangle.
Solution
We are given: 

sides  angles 
a = 20  A = 55.4° 
b = 24  B = ? 
c = ?  C = ? 

Since sin is positive in the 1st and 2nd quadrants, then B could also be 180°  81.03° = 98.97°.
Now we have:
sides  angles 
a = 20  A = 55.4° 
b = 24  B = 81.03° 
c = ?  C = 43.57° 


sides  angles 
a = 20  A = 55.4° 
b = 24  B = 98.97° 
c = ?  C = 25.63° 


 
C = 180°  55.4°  81.03° = 43.57° 

C = 180°  55.4°  98.97° = 25.63° 
We now need to use the law of sines with C = 43.57° and C = 25.63°:
C sin 55.40° = 20 sin 43.57° 

C sin 55.40° = 20 sin 25.63° 



Finally we have:
sides  angles 
a = 20  A = 55.4° 
b = 24  B = 81.03° 
c = 16.75  C = 43.57° 


sides  angles 
a = 20  A = 55.4° 
b = 24  B = 98.97° 
c = 10.51  C = 25.63° 

Note:
If sin is larger than 1, then the triangle is impossible.
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