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5.6 The Law of Sines How do we solve triangles that are not right triangles? B a c A C b Dec 212:20 PM B a c A b C The Law of Sines: Let ΔABC be any triangle with a, b and c representing the measures of the sides opposite the angles with measures A, B and C, respectively. Then, the following is true. a = sinA b sinB = c sinC Jan 33:17 PM 1 Examples: Solve each triangle. Round to the nearest tenth. 1.) B = 100o , C = 50o , c = 30 Dec 212:30 PM 2.) c = 19.3 , A = 39o15' , C = 64o45' Dec 212:36 PM 2 Area of Triangles Let ΔABC be any triangle. Then the area K can be determined by using one of the following formulas: K = ½bc sinA K = ½ab sinC K = ½ac sinB Note: Use these when given two sides and one angle. K = ½a2sinB sinC sinA K = ½c2 sinA sinB sinC K = ½b2 sinA sinC sinB Note: Use these when given one side and two angles. Dec 212:43 PM Examples: Find the area of each triangle. Round to the nearest tenth. 1.) a = 5, B = 37o , C = 84o 2.) b = 146.2, c = 209.3 , A = 62.2o Dec 212:50 PM 3 Homework Hint! Problem #31, page 317: A hot air balloon is flying above Groveburg. To the left side of the balloon, the balloonist measures the angle of depression to the Groveburg soccer fields to be 20o15'. To the right side of the balloon, the balloonist measures the angle of depression to the high school football field to be 62o30'. The distance between the two athletic complexes is 4 miles. Set up the picture! a b h x 4-x Jan 32:57 PM 5.6 ICE You may work together, but each person must submit their own work. 1.) Given A = 65o , B = 50o and c = 12: a.) Solve the triangle. Round to the nearest tenth. b.) Find the area of the triangle. Round to the nearest tenth. 2.) Challenge (Bonus): Derive the Law of Sines from the B following figure: c A a h C b Dec 212:55 PM 4