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**Law of Sines The Ambiguous Case**

Section 6-1

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**… this means we can solve them! Angle Side Angle - ASA **

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)

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**Side Side Angle - SSA There’s one left …**

There’s a problem with solving triangles given SSA … You could find… No solution One solution Two solutions In other words … its AMBIGUOUS … unclear Let’s take a look at each of these possibilities.

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**Remember now … the information we’re given is two consecutive sides and the next angle …**

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. b a θ

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**BOTH! What if side a is a little too long … what would that look like?**

a > b sin θ 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 a b θ B

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**Lastly, solve the obtuse triangle …**

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. a b θ B’ B

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**What if side a is larger than side b?**

Here’s the last scenario while θ is acute … What if side a is larger than side b? 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. a θ

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Laws of Sines. Introduction In the last module we studied techniques for solving RIGHT triangles. In this section and the next, you will solve OBLIQUE.

Laws of Sines. Introduction In the last module we studied techniques for solving RIGHT triangles. In this section and the next, you will solve OBLIQUE.

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