6 2 LAW OF COSINES What You Should

6. 2 LAW OF COSINES

What You Should Learn • Use the Law of Cosines to solve oblique triangles (SSS or SAS). • Use the Law of Cosines to model and solve real-life problems. • Use Heron’s Area Formula to find the area of a triangle. 2

Introduction Two cases remain in the list of conditions needed to solve an oblique triangle—SSS and SAS. If you are given three sides (SSS), or two sides and their included angle (SAS), none of the ratios in the Law of Sines would be complete. In such cases, you can use the Law of Cosines. 3

Let's consider types of triangles with the three pieces of information shown below. We can't use the Law of Sines on these because we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles. SAS You may have a side, an angle, and then another side SSS You may have all three sides AAA You may have all three angles. This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down"

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 -5 8. 2 -5

Proof of the Law of Cosines B a C x c h D b-x b Prove: c 2 = a 2 + b 2 – 2 ab cos C A Prove that c 2= a 2 + b 2 – 2 ab cos C In triangle CBD, cos C = x / a Then, x= a cos C (Eq #1) Using Pythagorean Theorem h 2 = a 2 - x 2 (Eq #2) In triangle BDA, Using Pythagorean Theorem c 2 = h 2 + (b – x)2 c 2 = h 2 + b 2 – 2 bx + x 2 (Eq #3) Substitute with h 2 Eq #2 c 2 = a 2 - x 2 + b 2 – 2 bx + x 2 Combine like Terms c 2 = a 2 - x 2 + b 2 – 2 bx + x 2 c 2 = a 2 + b 2 – 2 bx Finally, substitute x with Eq #1 c 2 = a 2 + b 2 – 2 ba cos C Therefore: c 2 = a 2 + b 2 – 2 ab cos C

Introduction 7

Using the Law of Cosines to Solve a Triangle (SAS) Example Solve triangle ABC if A = 42. 3°, b = 12. 9 meters, and c = 15. 4 meters. Copyright © 2007 Pearson Education, Inc. Slide 10 -8

Using the Law of Cosines to Solve a Triangle (SAS) Angle B must be the smaller of the two remaining angles since it is opposite the shorter of the two sides b and c. Therefore, it cannot be obtuse. We use the Law of Sines to find angle B. 10. 47 m Caution If we had chosen to find angle C rather than angle B, we would not have known whether angle C equals 81. 7° or its supplement, 98. 3°. Copyright © 2007 Pearson Education, Inc. Slide 10 -9

Law of Cosines Knowing the cosine of an angle, you can determine whether the angle is acute or obtuse. That is, cos > 0 for 0 < < 90 Acute cos < 0 for 90 < < 180. Obtuse If the largest angle is acute, the remaining two angles are acute also. 10

Using the Law of Cosines to Solve a Triangle (SAS) Step 1: Use the Law of Cosines to find the side opposite the given angle. Step 2: Use the Law of Sines to find the angle opposite the shorter of the two given sides. This angle is always acute. Step 3: Find the third angle. Subtract the measure of the given angle and the angle found in step 2 from 180°. Copyright © 2007 Pearson Education, Inc. Slide 10 -11

Using the Law of Cosines to Solve a Triangle (SSS) Example Solve triangle ABC if a = 9. 47 ft, b =15. 9 ft, and c = 21. 1 ft. Solution We solve for C, the largest angle, first. If cos C < 0, then C will be obtuse. C B Copyright © 2007 Pearson Education, Inc. A Slide 10 -12

Using the Law of Cosines to Solve a Triangle (SSS) We will use the Law of Sines to find B. C B Copyright © 2007 Pearson Education, Inc. A Slide 10 -13

Using the Law of Cosines to Solve a Triangle (SSS) • Use the Law of Cosines to find the angle opposite the longest side. • Use the Law of Sines to find either of the two remaining acute angles. • Find the third angle by subtracting the measures of the angles found in steps 1 and 2 from 180°. Copyright © 2007 Pearson Education, Inc. Slide 10 -14

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°. Find the distance AB. Copyright © 2009 Pearson Addison-Wesley 1. 1 -15 8. 2 -15

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 627 m. Copyright © 2009 Pearson Addison-Wesley 1. 1 -16 8. 2 -16

Example 3 – An Application of the Law of Cosines The pitcher’s mound on a women’s softball field is 43 feet from home plate and the distance between the bases is 60 feet, as shown in Figure 6. 13. (The pitcher’s mound is not halfway between home plate and second base. ) How far is the pitcher’s mound from first base? Figure 6. 13 17

Example 3 – Solution In triangle HPF, H = 45 (line HP bisects the right angle at H), f = 43, and p = 60. Using the Law of Cosines for this SAS case, you have h 2 = f 2 + p 2 – 2 fp cos H = 432 + 602 – 2(43)(60) cos 45 1800. 3. So, the approximate distance from the pitcher’s mound to first base is 42. 43 feet. 18

Area Formula • The law of cosines can be used to derive a formula for the area of a triangle given the lengths of three sides known as Heron’s Formula If a triangle has sides of lengths a, b, and c and if the semiperimeter is Then the area of the triangle is Copyright © 2007 Pearson Education, Inc. Slide 10 -19

Example 5 – Using Heron’s Area Formula Find the area of a triangle having sides of lengths a = 43 meters, b = 53 meters, and c = 72 meters. Solution: Because s = (a + b + c)/2 = 168/2 = 84, Heron’s Area Formula yields 1131. 89 square meters. 20

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 -21 8. 2 -21

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 -22 8. 2 -22

When to use the Law of Sines and the Law of Cosines Four possible cases can occur when solving an oblique triangle. Copyright © 2009 Pearson Addison-Wesley 1. 1 -23 8. 2 -23

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