CalculateMA 15400 Lesson 18 Section 8.1The Law of SinesAny triangle without a right angle is called an oblique triangle. As usual, we use , , for the angles of the triangle (or the vertices A, B, and C) and a, b, and c, respectively, for the lengths of the sides opposite those angles.The Law of Sines: In any triangle, the ratio of the sine of an angle to the side opposite that angle is equal to the ratio of the sine of another angle to the side opposite that angle.The Law of Sines:If ABC is an oblique triangle labeled in the usual manner, then:sinasinbsincSolve ABC : (This means given 3 pieces of data about a triangle, find the remaining 3 pieces of data.) = 41, = 77, a = 10.5a 10.5bca41.0°bg77.0° = 81, c = 11, b = 121abgabcAlways draw a triangle.This triangle does not exist. The side opposite the 81° angle must be longer than 11.ABCMA 15400 Lesson 18 Section 8.1The Law of SinesDraw Triangle:Review:Calculatesin(10) and sin(170) sin(80) and sin(100)sin(30) and sin(150) sin(1) and sin(179)Notice that the sines of supplementary angles are equivalent.Find the angle:sin(x) = 0.3456 sin(x) = 0.9876There are two angles that make each equation a true statement, one acute and one obtuse. This may or may not lead to two different triangles.When you are given one angle and two sides, and inverse sine is used to calculate an angle, there is a possibility of two triangles existing for the given information. This happens when there is SSA. Remember that the sum of the angles in a triangle is 180. = 53, a = 140, c = 115 Draw Δ1: Draw Δ 2:2MA 15400 Lesson 18 Section 8.1The Law of Sinesa 140.0 140.0bc 115.0 115.0ABC 53.0° 53.0°sin 53.0 sin115 140.0140(sin 53 )sin115sin 0.97225192576.47 180 76.47 103.53Since 103.5 + 53 is less than 180 degrees, there are two different triangles.AAAA A==== = - =ooo o3MA 15400 Lesson 18 Section 8.1The Law of Sines = 75, b = 248, c =195 Draw Δ:ab 248.0c 195.0AB 75.0°Csin 75.0 sin248.0 195.0195(sin 75 )sin248sin 0.7594981349.42 180 49.42 130.58However, 130.6 + 75.0 is greater than 180degrees. There will not be two distinct triangles here.CCCC C==== = - =ooo o4Final Answers: The largest side should always be opposite the largest angle. The smallest side should always be opposite the smallest angle.MA 15400 Lesson 18 Section 8.1The Law of SinesFire lookout: A fire is spotted from two fire lookouts stations. If station B reports the fire at S37E, and station A, 10 miles due East of station B, reports the fire at S61W, how far is the fire from station A? How far is the fire from station B?From Point B 4.8 MilesFrom Point A 8.1 Miles5Notice: The angles given for the measures are outside of the triangle formed from baMA 15400 Lesson 18 Section 8.1The Law of SinesFinding the Height of an Airplane. An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes between them and over the line joining them, observer A sights a 40 angle of elevation to the plane, and observer B sights a 35 angle. How high is the airplane?6There are 3 triangles in this picture,the big one and two right triangles.MA 15400 Lesson 18 Section 8.1The Law of SinesFinding the Length of a Ski Lift. a) What is the length of the ski lift? b) What is the height of the mountain?sin10 sin15sin 251000MountainSki Lift Ski Lift= =o oo725151000 ftADBCMountainSki LiftAgain, there are3 triangles, the big one and twosmaller ones, a left one and a right. In the left triangle, angle A would be 155° and the very small angle up by B is10°.MA 15400 Lesson 18 Section 8.1The Law of SinesSurveying A surveyor notes that the direction from point A to point B is N78E and the direction from A to C is N21E. The distance from A to B is 345 meters and the distance from B to C is 543 meters. Approximate the distance from A to C.8Note: This picture may notbe to
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