Copyright © 2007 Pearson Education, Inc. Slide 10-1 While you wait: Without consulting any resources or asking your friends… write down everthing you remember about the:

Copyright © 2007 Pearson Education, Inc. Slide 10-2 Sec 9.3 The Law of Sines Oblique Triangles

Copyright © 2007 Pearson Education, Inc. Slide 10-3 Recall In a triangle, the sum of the interior angles is 180º. No triangle can have two obtuse angles. The height of a triangle is less than or equal to the length of two of the sides..

Copyright © 2007 Pearson Education, Inc. Slide 10-4 Recall The sine function has a range of If the θ is a positive decimal < 1, the θ can lie in the first quadrant (acute <) or in the second quadrant (obtuse <)..

Copyright © 2007 Pearson Education, Inc. Slide Data Required for Solving Oblique Triangles Case 1One side and two angles known: »SAA or ASA Case 2Two sides and one angle not included between the sides known: »SSA »This case may lead to more than one solution. Case 3Two sides and one angle included between the sides known: »SAS Case 4Three sides are known: »SSS

Copyright © 2007 Pearson Education, Inc. Slide Derivation of the Law of Sines For our purposes we will use the triangles labelled as : The orientation of the triangle will not make a difference.

Copyright © 2007 Pearson Education, Inc. Slide Derivation of the Law of Sines Start with an acute or obtuse triangle and construct the perpendicular from B to side AC. Let h be the height of this perpendicular. Then c and a are the hypotenuses of right triangle ADB and BDC, respectively.

Copyright © 2007 Pearson Education, Inc. Slide The Law of Sines In a similar way, by constructing perpendiculars from other vertices, the following theorem can be proven. The Law of Sine Use to find ANGLESUse to find SIDES

Copyright © 2007 Pearson Education, Inc. Slide 10-9 Yet another version of the Law of Sines

Copyright © 2007 Pearson Education, Inc. Slide Using the Law of Sines to Solve a Triangle ExampleSolve triangle ABC if A = 32.0°, B = 81.8°, and a = 42.9 centimeters. SolutionDraw the triangle and label the known values. Because A, B, and a are known, we can apply the law of sines involving these variables.

Copyright © 2007 Pearson Education, Inc. Slide Using the Law of Sines to Solve a Triangle To find C, use the fact that there are 180° in a triangle. Now we can find c

Copyright © 2007 Pearson Education, Inc. Slide Law of Sines Day 2 For each given question 1-9: Draw a triangle with proper labels and place the given values on the diagram. Identify the combination given i.e. SAS, etc. We will be using these diagrams to solve each triangle and to record notes… so allow room around each shape for further writing.

Copyright © 2007 Pearson Education, Inc. Slide Example

Copyright © 2007 Pearson Education, Inc. Slide For each diagram: Identify whether L.O.S. can be used. If LOS can be used: How many solutions:0, 1 or 2? Can you make some generalization; or write a program? As you make any generalization, update your triangle table.

Copyright © 2007 Pearson Education, Inc. Slide 10-15

Copyright © 2007 Pearson Education, Inc. Slide Repeat the process with the following

Copyright © 2007 Pearson Education, Inc. Slide Using the Law of Sines in an Application (ASA) ExampleTwo stations are on an east-west line 110 miles apart. A forest fire is located on a bearing of N 42° E from the western station at A and a bearing of N 15° E from the eastern station at B. How far is the fire from the western station? SolutionAngle BAC = 90° – 42° = 48° Angle B = 90° + 15° = 105° Angle C = 180° – 105° – 48° = 27° Using the law of sines to find b gives

Copyright © 2007 Pearson Education, Inc. Slide Ambiguous Case If given the lengths of two sides and the angle opposite one of them, it is possible that 0, 1, or 2 such triangles exist. Some basic facts that should be kept in mind: –For any angle , –1  sin   1, if sin  = 1, then  = 90° and the triangle is a right triangle. –sin  = sin(180° –  ). –The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-value angle is opposite the intermediate side (assuming unequal sides).

Copyright © 2007 Pearson Education, Inc. Slide Number of Triangles Satisfying the Ambiguous Case Let sides a and b and angle A be given in triangle ABC. (The law of sines can be used to calculate sin B.) 1.If sin B > 1, then no triangle satisfies the given conditions. 2.If sin B = 1, then one triangle satisfies the given conditions and B = 90°. 3.If 0 < sin B < 1, then either one or two triangles satisfy the given conditions (a)If sin B = k, then let B 1 = sin -1 k and use B 1 for B in the first triangle. b)Let B 2 = 180° – B 1. If A + B 2 < 180°, then a second triangle exists. In this case, use B 2 for B in the second triangle.

Copyright © 2007 Pearson Education, Inc. Slide Ambiguous Case a < b sinA a = b sinA a < b sinA a > b sinA

Copyright © 2007 Pearson Education, Inc. Slide How does it work? Web demo

Copyright © 2007 Pearson Education, Inc. Slide Ambiguous Case for Obtuse Angle A

Copyright © 2007 Pearson Education, Inc. Slide Solving the Ambiguous Case: One Triangle Example Solve the triangle ABC, given A = 43.5°, a = 10.7 inches, and c = 7.2 inches. Solution The other possible value for C: C = 180° – 27.6° = 152.4°. Add this to A: 152.4° ° = 195.9° > 180° Therefore, there can be only one triangle.

Copyright © 2007 Pearson Education, Inc. Slide Solving the Ambiguous Case: One Triangle

Copyright © 2007 Pearson Education, Inc. Slide Solving the Ambiguous Case: No Such Triangle ExampleSolve the triangle ABC if B = 55°40´, b = 8.94 meters, and a = 25.1 meters. SolutionUse the law of sines to find A. Since sin A cannot be greater than 1, the triangle does not exist.

Copyright © 2007 Pearson Education, Inc. Slide Example Solve the triangle ABC if A = 55.3°, a = 22.8 feet, and b = 24.9 feet. Solution 9.3Solving the Ambiguous Case: Two Triangles

Copyright © 2007 Pearson Education, Inc. Slide Solving the Ambiguous Case: Two Triangles To see if B 2 = 116.1° is a valid possibility, add 116.1° to the measure of A: 116.1° ° = 171.4°. Since this sum is less than 180°, it is a valid triangle. Now separate the triangles into two: AB 1 C 1 and AB 2 C 2.

Copyright © 2007 Pearson Education, Inc. Slide Solving the Ambiguous Case: Two Triangles Now solve for triangle AB 2 C 2.

Copyright © 2007 Pearson Education, Inc. Slide Practice: Answer in pairs. Find m  B, m  C, and c, if they exist. 1) a = 9.1, b = 12, m  A = 35 o 2) a = 25, b = 46, m  A = 37 o 3) a = 15, b = 10, m  A = 66 o

Copyright © 2007 Pearson Education, Inc. Slide Answers: 1)Case 1: m  B=49 o,m  C=96 o,c=15.78 Case 2: m  B=131 o,m  C=14 o,c=3.84 2)No possible solution. 3)m  B=38 o,m  C=76 o,c=15.93

Copyright © 2007 Pearson Education, Inc. Slide Practice Problems on your own: The problems below is a sampling of what you should try; feel free to try other problems also: Sec 9-3; Written Exercises, page 347 #8,14, all