1. Essential Question: What is the law of sines, and how do we apply it?

2. 10-2: The Law of Sines For these next two sections, you will need to be in degree mode In any triangle ABC (in standard notation), the law of sines states Note: If you’re using the law of sines to find an angle, it will be easier if you flip all of the ratios above (Proof on board)

3. 10-2: The Law of Sines Solve a Triangle with AAS Information Example 1: If B = 20 °, C = 31 ° and b = 210, find the other angle measure and side lengths. Sometimes it helps if you draw a triangle Label it Finding A should be obvious Use the law of sines to find the side lengths (next slide) A B C 20 ° 31 ° 210 129 °

4. 10-2: The Law of Sines Solve a Triangle with AAS Information Use the law of sines to find the side lengths A B C 20 ° 31 ° 210 129 °

5. 10-2: The Law of Sines The ambiguous case When dealing with an AAS triangle, there’s only one solution – that goes back to your rules about triangle similarity (the same is true when given ASA and SAS) However, when dealing with a triangle with SSA (or ASS) information, we’re left with some unknowns… ? ?  A

6. 10-2: The Law of Sines Solving a Triangle with SSA Information (no solution) Example 2: Given a possible triangle ABC with a = 6, b = 7 and A = 65 °, find angle B. Use the law of sines (flip the law – make your life easier) Because the maximum of a sine function is 1, there is no B possible, and there is no triangle possible.

7. 10-2: The Law of Sines Solving a Triangle with SSA Information (one solution) Example 3: An airplane A takes off from carrier B and flies in a straight line for 12 km. At that instant, an observer on destroyer C, located 5 km from the carrier, notes that the angle determined by the carrier, the destroyer (vertex) and the plane is 37 °. How far is the plane from the destroyer? A B C 37 ° 5 12 (not possible)

8. 10-2: The Law of Sines Solving a Triangle with SSA Information (two solutions) Example 4:Solve triangle ABC when a = 7.5, b = 12, and A = 35 °. (continued next slide) A B C 35 ° 7.5 12

9. 10-2: The Law of Sines Solving a Triangle with SSA Information (two solutions) Example 4:Solve triangle ABC when a = 7.5, b = 12, and A = 35 °. Case 1: B = 66.6 ° Case 2: B = 113.4 ° C = 180 – 35 – 66.6 = 78.4 ° C = 180 – 35 – 113.4 = 31.6 °

10. 10-2: The Law of Sines Assignment Page 634 1 – 7 17 – 25 33 – 35 odd problems Show work