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

Applications of Trigonometry and Vectors 7 7.1 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 7.5 Applications of Vectors Copyright © 2009 Pearson Addison-Wesley

7.3 The Law of Cosines Derivation of the Law of Cosines ▪ Solving SAS and SSS Triangles (Cases 3 and 4) ▪ Heron’s Formula for the Area of a Triangle Copyright © 2009 Pearson Addison-Wesley 1.1-3

Triangle Side Length Restriction In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Copyright © 2009 Pearson Addison-Wesley 1.1-4

Derivation of the Law of Cosines Let ABC be any oblique triangle located on a coordinate system as shown. The coordinates of A are (x, y). For angle B, and Thus, the coordinates of A become (c cos B, c sin B). Copyright © 2009 Pearson Addison-Wesley

Derivation of the Law of Cosines (continued) The coordinates of C are (a, 0) and the length of AC is b. Using the distance formula, we have Square both sides and expand. Copyright © 2009 Pearson Addison-Wesley

In any triangle, with sides a, b, and c, Law of Cosines In any triangle, with sides a, b, and c, Copyright © 2009 Pearson Addison-Wesley 1.1-7

Note If C = 90°, then cos C = 0, and the formula becomes the Pythagorean theorem. Copyright © 2009 Pearson Addison-Wesley 1.1-8

Example 1 USING THE LAW OF COSINES IN AN APPLICATION (SAS) A surveyor wishes to find the distance between two inaccessible points A and B on opposite sides of a lake. While standing at point C, she finds that AC = 259 m, BC = 423 m, and angle ACB measures 132°40′. Find the distance AB. Copyright © 2009 Pearson Addison-Wesley 1.1-9

The distance between the two points is about 628 m. Example 1 USING THE LAW OF COSINES IN AN APPLICATION (SAS) Use the law of cosines because we know the lengths of two sides of the triangle and the measure of the included angle. The distance between the two points is about 628 m. Copyright © 2009 Pearson Addison-Wesley 1.1-10

Solve triangle ABC if A = 42.3°, b = 12.9 m, and c = 15.4 m. Example 2 USING THE LAW OF COSINES TO SOLVE A TRIANGLE (SAS) Solve triangle ABC if A = 42.3°, b = 12.9 m, and c = 15.4 m. B < C since it is opposite the shorter of the two sides b and c. Therefore, B cannot be obtuse. Copyright © 2009 Pearson Addison-Wesley 1.1-11

Use the law of sines to find the measure of another angle. Example 2 USING THE LAW OF COSINES TO SOLVE A TRIANGLE (SAS) (continued) Use the law of sines to find the measure of another angle. ≈ 10.47 Now find the measure of the third angle. Copyright © 2009 Pearson Addison-Wesley 1.1-12

Caution If we used the law of sines to find C rather than B, we would not have known whether C is equal to 81.7° or its supplement, 98.3°. Copyright © 2009 Pearson Addison-Wesley 1.1-13

Solve triangle ABC if a = 9.47 ft, b = 15.9 ft, and c = 21.1 ft. Example 3 USING THE LAW OF COSINES TO SOLVE A TRIANGLE (SSS) Solve triangle ABC if a = 9.47 ft, b = 15.9 ft, and c = 21.1 ft. Use the law of cosines to find the measure of the largest angle, C. If cos C < 0, angle C is obtuse. Solve for cos C. Copyright © 2009 Pearson Addison-Wesley 1.1-14

Now find the measure of angle A. Example 3 USING THE LAW OF COSINES TO SOLVE A TRIANGLE (SSS) (continued) Use either the law of sines or the law of cosines to find the measure of angle B. Now find the measure of angle A. Copyright © 2009 Pearson Addison-Wesley 1.1-15

Find the measure of angle B in the figure. Example 4 DESIGNING A ROOF TRUSS (SSS) Find the measure of angle B in the figure. Copyright © 2009 Pearson Addison-Wesley 1.1-16

Four possible cases can occur when solving an oblique triangle. Copyright © 2009 Pearson Addison-Wesley 1.1-17

Copyright © 2009 Pearson Addison-Wesley 1.1-18

Heron’s Area Formula (SSS) If a triangle has sides of lengths a, b, and c, with semiperimeter then the area of the triangle is Copyright © 2009 Pearson Addison-Wesley 1.1-19

Example 5 USING HERON’S FORMULA TO FIND AN AREA (SSS) The distance “as the crow flies” from Los Angeles to New York is 2451 miles, from New York to Montreal is 331 miles, and from Montreal to Los Angeles is 2427 miles. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of Earth.) Copyright © 2009 Pearson Addison-Wesley 1.1-20

Using Heron’s formula, the area  is Example 5 USING HERON’S FORMULA TO FIND AN AREA (SSS) (continued) The semiperimeter s is Using Heron’s formula, the area  is Copyright © 2009 Pearson Addison-Wesley 1.1-21