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# Class Note for MATH 1330 at UH 2

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Date Created: 02/06/15

Math 1330 Section 73 The Law of Sines and the Law of Cosines Sometimes you will need to solve a triangle that is not a right triangle This type of triangle is called an oblique triangle To solve an oblique triangle you will not be able to use right triangle trigonometry Instead you will use the Law of Sines and or the Law of Cosines You will typically be given three parts of the triangle and you will be asked to nd the other three The approach you will take to the problem will depend on the information that is given If you are given SSS the lengths of all three sides or SAS the lengths of two sides and the measure of the included angle you will use the Law of Cosines to solve the triangle If you are given SAA the measures of two angles and one side or SSA the measures of two sides and the measure of an angle that is not the included angle you will use the Law of Sines to solve the triangle Recall from your geometry course that SSA does not necessarily determine a triangle We will need to take special care when this is the given information Here s the Law of Cosines In any triangle ABC a2 b2 c2 2bccosA b2 a2 c2 2accosB c2 a2 b2 2abcosC The development of one case of this formula is given in detail in the online text Here s the Law of Sines In any triangle ABC sinA sinB sinC a b c The development of this formula is given in detail in the online text Here are some facts about solving triangles that may be helpful in this section If you are given SSS SAS or SAA the information determines a unique triangle If you are given SSA the information given may determine 0 l or 2 triangles If this is the information you are given you will have some additional work to do Since you will have three pieces of information to nd when solving a triangle it is possible for you to use both the Law of Sines and the Law of Cosines in the same problem When drawing a triangle the measure of the largest angle is opposite the longest side the measure of the middlesized angle is opposite the middlesized side and the measure of the smallest angle is opposite the shortest side Suppose a b and c are suggested to be the lengths ofthe three sides ofa triangle Suppose that c is the biggest of the three measures In order for a b and c to form a triangle this inequality must be true a b gt c So the sum of the two smaller sides must be greater than the third side An obtuse triangle is a triangle which has one angle that is greater than 90 An acute triangle is a triangle in which all three angles measure less than 90 If you are given the lengths of the three sides of a triangle where c gt a and c gt b you can determine if the triangle is obtuse or acute using the following If a2 b2 gt c2 the triangle is an acute triangle If a2 b2 lt cl the triangle is an obtuse triangle Your first task will be to analyze the given information to determine which formula to use You should sketch the triangle and label it with the given information to help you see what you need to find If you have a choice it is usually best to find the largest angle first Example 1 In AABC a 6 b 8 and c 12 Find the measures ofthe three angles to the nearest tenth of a degree Example 2 In AABC AB 51quot a 14 and c 10 Solve the triangle Round all answers to the nearest hundredth Example 3 InAPQRp 4 q 7 and r 13 Solve the triangle Round all answers to the nearest hundredth Example 4 In AXYZ AX 24quot AZ 840 and y 18 Solve the triangle and round all answers to the nearest hundredth Example 5 In AABC AA 43 b 9 and a 6 Solve the triangle and round all answers to the nearest hundredth Example 6 In APQR AP 20 p 30 and q 36 Solve the triangle and round all answers to the nearest hundredth Example 7 In AXYZ AY 22quot y 7 x 5 Solve the triangle and round all answers to the nearest hundredth Example 8 Solve the triangle and round all answers to the nearest hundredth 10 8 Example 9 Find the area of the quadrilateral 23 m 77 38m 19m

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