Law of Sines AAS ONE SOLUTION SSA AMBIGUOUS CASE ASA ONE SOLUTION Domain error NO SOLUTION Second angle option violates triangle angle-sum theorem ONE SOLUTION Both angles satisfy triangle angle-sum theorem TWO SOLUTIONS

Law of Cosines SAS Use LoC to solve for side corresponding to included angle first. Then use LoS to solve for smallest of remaining two angles. Find third through subtraction. SSS Use LoC to solve for largest angle first. Then use LoS to find either of remaining two angles. Find third through subtraction.

Example 1: a = 40u, c = 44u,  B = 16  A C B 40u 44u 16  Step 1: Identify triangle as SAS and therefore a Law of Cosines problem. Step 2: Use Law of Cosines to find side b which corresponds to the given included  B. Step 3: Use Law of Sines to find smallest angle. Why? Law of Sines requires use of inverse sin which can only yield an acute angle. Side c is largest side therefore  C is largest angle which can be obtuse or acute and Law of Sines will not distinguish. Since a triangle can only have one obtuse angle and that angle would have to be the largest angle, then  A must be acute which can be solved using Law of Sines. Step 4: Find the third angle using subtraction and the triangle angle-sum theorem. OBTUSE!

Example 2: a = 18u, b = 10u, c = 25u A C B 18u 25u 10u Step 1: Identify triangle as SSS and therefore a Law of Cosines problem. Step 2: Use Law of Cosines to find largest angle first. Why? Again, a triangle can only have one obtuse angle and when you use inverse cosine, the answer will determine acute or obtuse. Then you can switch to find the acute angles using Law of Sines. The largest side is c which corresponds to the largest angle  C. Step 3: Use Law of Sines to find either of the other two angles. Step 4: Find the third angle using subtraction and the triangle angle-sum theorem. OBTUSE!