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Trigonometry: Oblique Triangles - The Ambiguous Case
Before proceeding with this lesson, you should review the introductory lesson on the Law of Sines. The Law of Sines is used to find angle and side measurements for triangles where the givens fit in the cases of AAS or ASA. The Law of Sines can also be used in the SSA case, however, additional work is needed to verify the number of possible triangles that can result from being given this combination.
 
AAS ASA SSA

2 angles & non-included side.

2 angles & included side.

2 sides & non-included angle.
 
We will continue to follow the steps outlined in the previous lesson, with the addition of the steps needed for the Ambiguous Case.
 
  1. Draw a picture of the triangle and label all knowns.
  2. Do I have AAS or ASA?
 
  1. If yes, you can set up the Law of Sines and solve.
  2. If no, you may not be able to use the Law of Sines go to step 3.
 
  1. Do I have SSA?
 
  1. If no, have you been given SSS or SAS? If so, you need to use the Law of Cosines.
  2. If yes, this is an example of the Ambiguous Case. You must find the height of the triangle.
 
h = (side adjacent to given angle) sin (given angle)
 
  1. If h = (opp side) or if (opp side) > (adj side), then only one triangle is possible.
Set up the Law of Sines and solve for the unknowns.
 
**If side a is less than side b…check the following two cases.
 
  1. If h > (opp side), then no triangle is possible
 
In the following triangle, angle A, sides a and b are given. Side A is referred to as the “opposite” side because it is the side opposite the given angle. The perpendicular height of the triangle is the smallest distance needed to reach the base, therefore, it is the smallest distance needed to make a triangle. If side a is not even as long as the height, it will not reach the other side of the triangle, and no triangle is possible.
 
 
  1. If h < (opp side) < (adj side), then two triangles are possible
 
In the following triangle, angle A, sides a and b are given. Side A is referred to as the “opposite” side because it is opposite the given angle. The perpendicular height of the triangle is smaller than side a, so side a will reach the base of the triangle. In addition, both the height and side a are smaller than side b, so that it is possible to create two different triangles with a given length for side a:
 
and obtuse triangle (marked with side a1)
and an acute triangle (marked with side a2)
 
 
You must set up a Law of Sines ratio to find the reference angle, or θ, which is the angle at which side a meets the base of the triangle. Then proceed as follows:
 
  • find θ’s:
  • solve both triangles

 
Let's practice solving some triangles!
ExamplesExamples:
Solve the triangle: A = 58º, a = 20, b = 10
 

Solve the triangle: a = 12 meters, b = 17 meters, A = 71°
 

Solve the triangle: A = 35°, a = 133, b = 230
 




K Dodd
M Dionne

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