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The Law of Sines and fishing lines In the remote village of Seward, Alaska there is a five year waiting list for docking space for fishing boats. Law of Sines Objectives: • I will be able to use the Law of Sines to solve Triangles. • I will be able to solve problems using the Law of Sines. Key – The Law of Sines can be used to find missing parts of triangles that are NOT right triangles. The Law of Sines The Law of Sines can be used in the following two cases: 1. We know the measures of two angles and any side of a triangle. 2. You know two sides and the measure of one angle opposite one of these sides. The Law of Sines Let ABC be any triangle with a, b, and c being the sides opposite the angles A, B, and C. C a b A Then: c sin A sin B sin C a b c B Depending on the type of fishing that you are doing, it is important to determine how much fishing line you need A 5-foot fishing pole is anchored to the edge of a dock in Seward, Alaska. If the distance from the foot of the pole to the point where the fishing line meets the water is 45 feet, about how much fishing line that is cast out is above the surface of the water? Use: sin A sin B sin C a b c We are looking for side a. sin 54 sin120 a 45 45(sin54) = a(sin120) 45(sin 54) a sin120 Answer: About 42 feet of the fishing line that is cast out is above the surface of the water. The Law of Sines - Alternate Basic Form Alternate Form: sin A sin B sin C a b c a b c sin A sin B sin C Easier to use when trying to find a side. A direct flight from New York to Anchorage is a shorter than flying with a layover in Chicago. With the given distances below, figure out how much shorter a direct flight would be. Use the Law of Sines to figure out the distance from Chicago to Anchorage. Round to the 10ths place. Work with a partner, discuss and compare answers. Anchorage x 2° 3,360 mi. New York .5° x x 725 mi. Chicago Answer: About 264 miles Example 1 Find side p. Round to the nearest tenth. Answer: Example 2 to the nearest degree in Law of Sines Cross products Solve for L. Answer: , Solving a Triangle The Law of Sines can be used to solve a triangle. Solving a triangle means finding the measures of all the sides and all the angles of a triangle. You may also need to use the angle sum theorem to solve the triangle. This is how we solve for distances and angles for remote mountain ranges. Example 3 . Round angle measures to the nearest degree and side measures to the nearest tenth. To To find find e: d: 8 60 112 We know the measures of two angles of the triangle. Use the Angle Sum Theorem to find Answer: After a tremendous earthquake struck Alaska in the mid 1960’s villages were destroyed and power lines were broken or in the state of disrepair. The following problem was used to determine the acceptable limits of bent and damaged telephone poles Example 3 A 46-foot telephone pole tilted at an angle of from the vertical casts a shadow on the ground. Find the length of the shadow to the nearest foot when the angle of elevation to the sun is Draw a diagram Draw Then find the Example 3 Since you know the measures of two angles of the triangle, and the length of a side opposite one of the angles you can use the Law of Sines to find the length of the shadow. Answer: The length of the shadow is about 75.9 feet. Your Turn Design one problem on your own – Start by thinking what you like to do the most • Ex. Play an instrument, sports, activity • How/where does it form a triangle – Next, add the dimensions for one of the following: • The measures of two angles and any side of a triangle. • The measures of two sides and the measure of one angle opposite one of these sides. – Share your sample problem with a neighbor and see if they can solve it. Homework • Make 6 additional problems tonight • 3 that provide the measures of two angles and any side of a triangle. • 3 that provide the measures of two sides and the measure of one angle opposite one of these sides. • All should relate to your own experiences… • Look out the window, there are examples all around.