Chapter 8G - Law of Sines and Law of Cosines

Transcription

1 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 246 Chapter 8G - Law of Sines and Law of Cosines Given a general triangle, labeled as below Two interesting truths exist: A. The Law of Sines!!! sin A a = sin B b = sinc c B. The Law of Cosines:!!! c 2 = a 2 + b 2 2abcosC Notice that the Pythagorean Theorem is a special case of the Law of Cosines! In a right triangle, C = so cosc = and we have

2 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 247 Understanding the possibilities! In geometry, we learned that if 2 triangles have corresponding sides of the same length, then they are congruent. It is usually referred to as SSS Congruence Theorem. So if we are given a set of three numbers that represent the lengths of the sides of a triangle, then there are two possibilities: 1. No triangle with those side lengths or 2. Exactly 1 triangle with those side lengths. If the triangle exists, then our goal will be to solve the triangle. That means we will want to determine the measures of the three angles. 1. If a = 3, b = 4, and c = 8, we know that no triangle exists because < 8 2. If a = 2, b = 3, and c = 4, then we could use the Law of Cosines to determine the measures of the 3 angles.

3 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 248 There was the SAS Congruence Theorem. It told us that if 2 sides and the included angle of one triangle are congruent to the corresponding sides of another triangle, then the 2 triangles are congruent. From this we know that if we are given the lengths of 2 sides of a triangle and the measure of the included angle, then there is exactly one triangle with these measurements. Our goal again will be to solve the triangle. This means we want to determine the length of the third side and the measures of the other two angles. 3. If a=14, b=16, and C = 120, determine the length of side c. Here, too, we would need to start with the Law of Cosines.

4 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 249 There were also the AAS and ASA Congruence Theorems. They told us that if 2 angles and one of the sides of a triangle was congruent to the corresponding parts of another triangle, then the triangles were congruent. This could be interpreted another way. If we are given 2 angle measures whose sum is less than 180 and the length of one of the sides of the triangle, then we can determine the angle measure of the third angle and the length of the other 2 sides of the triangle. Here the Law of Sines is the easiest to use. 4. If A = 75, C= 65, and a=3, solve the triangle.

5 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 250 Finally we get to the congruence theorem that does not exist: SSA. If you have 2 triangles, and even if you know that 2 sides on one triangle are congruent to 2 sides on the other triangle and one of the angles opposite one of the sides is congruent to the corresponding angle of the other triangle, then you cannot be sure that the triangles are congruent. For us this means that if we are given 2 numbers that represent the lengths of the sides of a triangle and an angle measure that is not between the given sides, then there are three possibilities for that set of numbers. 1. The given conditions might be such that no triangle exists. 2. The given conditions might be such that exactly 1 triangle exists. 3. The given conditions might be such that 2 different triangle exist. Page 3 of this document has nice diagrams that I think will help you understand %20Law%20of%20Sines.pdf 5. Given A = 75, a = 51, b = 71, determine B, C, and c -- if such a triangle exists..

6 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! Given.A = 37, a = 12, b = 16.1 determine B, C, and c -- if such a triangle exists..

7 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! Given A = 40, a = 20, b = 15 determine B, C, and c -- if such a triangle exists..

8 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! From ( %20Law%20of%20Sines.pdf) a)

9 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 254 b)

10 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 255 c)

11 Fry Texas A&M University Math 150 Chapter 8G Fall 2015! 256 Extra Problems:! Text:4-11!! 1. The sides of a triangle measure 3, 5, and 7. Determine the cosine of the largest angle. 2. Walking through Blocker, a classmate stops you. He is working on a problem from the section on Law of Sines and Law of Cosines. In the problem two side lengths and an angle measure of a triangle are given. Specifically, a = 30 units, b = 49 units, and A = 15!. Your classmate did some work with his calculator and determined (correctly) that B arcsin(.423) 25.0! Now he is unsure of what to do. Hopefully you can help. Circle and complete the best answer: (decimal approximations) I. This is a case in which no triangle with the given measurements exists. II. This is a case in which there is exactly one triangle B and C III This is a case in which there are exactly 2 triangles B and C or B and C IV This is the case with infinitely many triangles. V. None of the above. 3. sin( 18! ) = 0.3. If the measure of an obtuse angle is θ, and sinθ = 0.3, then θ = a) 0.7 b) 162! c) arcsinθ d) arccosθ e) None of these