Copyright © 2009 Pearson Addison-Wesley Applications of Trigonometry and Vectors

Copyright © 2009 Pearson Addison-Wesley Oblique Triangles and the Law of Sines 7.2 The Ambiguous Case of the Law of Sines 7.3 The Law of Cosines 7.4 Vectors, Operations, and the Dot Product 7.5Applications of Vectors 7 Applications of Trigonometry and Vectors

Copyright © 2009 Pearson Addison-Wesley The Ambiguous Case of the Law of Sines 7.2 Description of the Ambiguous Case ▪ Solving SSA Triangles (Case 2) ▪ Analyzing Data for Possible Number of Triangles

Copyright © 2009 Pearson Addison-Wesley Description of the Ambiguous Case If the lengths of two sides and the angle opposite one of them are given (Case 2, SSA), then zero, one, or two such triangles may exist.

Copyright © 2009 Pearson Addison-Wesley If A is acute, there are four possible outcomes.

Copyright © 2009 Pearson Addison-Wesley If A is obtuse, there are two possible outcomes.

Copyright © 2009 Pearson Addison-Wesley Applying the Law of Sines 1.For any angle θ of a triangle, 0 < sin θ ≤ 1. If sin θ = 1, then θ = 90° and the triangle is a right triangle. 2.sin θ = sin(180° – θ) (Supplementary angles have the same sine value.) 3.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 the triangle has sides that are all of different lengths).

Copyright © 2009 Pearson Addison-Wesley Example 1 SOLVING THE AMBIGUOUS CASE (NO SUCH TRIANGLE) Law of sines (alternative form) Solve triangle ABC if B = 55°40′, b = 8.94 m, and a = 25.1 m. Since sin A > 1 is impossible, no such triangle exists.

Copyright © 2009 Pearson Addison-Wesley Example 1 SOLVING THE AMBIGUOUS CASE (NO SUCH TRIANGLE) (continued) An attempt to sketch the triangle leads to this figure.

Copyright © 2009 Pearson Addison-Wesley Note In the ambiguous case, we are given two sides and an angle opposite one of the sides (SSA).

Copyright © 2009 Pearson Addison-Wesley Example 2 SOLVING THE AMBIGUOUS CASE (TWO TRIANGLES) Solve triangle ABC if A = 55.3°, a = 22.8 ft, and b = 24.9 ft. There are two angles between 0° and 180° such that sin B ≈ :

Copyright © 2009 Pearson Addison-Wesley Example 2 SOLVING THE AMBIGUOUS CASE (TWO TRIANGLES) (continued) Solve separately for triangles

Copyright © 2009 Pearson Addison-Wesley Example 2 SOLVING THE AMBIGUOUS CASE (TWO TRIANGLES) (continued)

Copyright © 2009 Pearson Addison-Wesley Example 2 SOLVING THE AMBIGUOUS CASE (TWO TRIANGLES) (continued)

Copyright © 2009 Pearson Addison-Wesley Number of Triangles Satisfying the Ambiguous Case (SSA) Let sides a and b and angle A be given in triangle ABC. (The law of sines can be used to calculate the value of sin B.) 1.If applying the law of sines results in an equation having 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°.

Copyright © 2009 Pearson Addison-Wesley Number of Triangles Satisfying the Ambiguous Case (SSA) 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 © 2009 Pearson Addison-Wesley Example 3 SOLVING THE AMBIGUOUS CASE (ONE TRIANGLE) Solve triangle ABC given A = 43.5°, a = 10.7 in., and c = 7.2 in. There is another angle C with sine value : C = 180° – 27.6° = 152.4°

Copyright © 2009 Pearson Addison-Wesley Example 3 SOLVING THE AMBIGUOUS CASE (ONE TRIANGLE) (continued) Since c < a, C must be less than A. So C = 152.4° is not possible. B = 180° – 27.6° – 43.5° = 108.9°

Copyright © 2009 Pearson Addison-Wesley Example 4 ANALYZING DATA INVOLVING AN OBTUSE ANGLE Without using the law of sines, explain why A = 104°, a = 26.8 m, and b = 31.3 m cannot be valid for a triangle ABC. Because A is an obtuse angle, it must be the largest angle of the triangle. Thus, a must be the longest side of the triangle. We are given that b > a, so no such triangle exists.