Download a > b sin θ

Law of Sines
The Ambiguous Case
Section 6-1
In geometry, triangles can be uniquely defined when particular
combinations of sides and angles are specified
… this means we can solve them!
Angle Side Angle - ASA
Angle Angle Side – AAS
We solved these using Law of Sines
Then there are these theorems …
Side Side Side – SSS
Side Angle Side – SAS
We’ll soon solve these using the Law of Cosines
(section 6.2)
There’s one left …
Side Side Angle - SSA
There’s a problem with solving triangles given SSA …
You could find…
1.
No solution
2.
One solution
3.
Two solutions
In other words … its AMBIGUOUS … unclear
Let’s take a look at each of these possibilities.
Remember now … the information we’re given is two consecutive sides
and the next angle …
b
a
θ
•
If this side isn’t long enough, then we can’t create a triangle … no solution
So, then, what is the “right” length so we can make a triangle?
An altitude … 90 degree angle … a RIGHT triangle!
Turns out this is an important calculation … it’s
a = b sin θ
•
•
•
If a = b sin θ, then there is only one solution for this triangle.
The missing angle is the complement to θ
The missing side can be found using Pythagorean theorem of
trigonometry.
What if side a is a little too long … what would that look like?
a > b sin θ
a
b
θ
B
This leg can then either swing left … or right.
So? … which one of these triangles do you solve? …
BOTH!
• First, solve the acute triangle … and find angle B by Law of Sines!
• Then solve for the remaining parts of the acute triangle
Lastly, solve the obtuse triangle …
a
b
θ
B’
B
This next step is critical … angle B’ is ALWAYS the supplement to angle
B.
B’ = 180 – m< B
• Next, solve the remaining parts of the obtuse triangle.
Here’s the last scenario while θ is acute …
What if side a is larger than side b?
a
θ
• In this case, only one triangle can exist … an acute triangle which
can easily be solved using law of sines.
Too long to create a triangle on this side.