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1. 5-Minute Check Lesson 5-8A
2. MathadorGameplan Section 5.8:The Law of Cosines CA Standards:T 13.0 Daily Objective (11/28/12): Students will be able to (1) solve triangles by using the Law of Cosines, and (2) find the area of triangles if the measures of the three sides are given (SSS). Homework:page 331 (#11 to 26 all)
3. Triangle Inequalities In any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
4. The coordinates of A are (x, y). For angle B, and Derivation of the Law of Cosines Let ABC be any oblique triangle located on a coordinate system as shown. Thus, the coordinates of A become (c cos B, c sin B).
5. 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.
6. Law of Cosines In any triangle, with sides a, b, and c,
7. Note If C = 90°, then cos C = 0, and the formula becomes the Pythagorean theorem.
8. 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.
9. Use the law of sines to find the measure of another angle. ≈ 10.47 Now find the measure of the third angle.
10. Solve triangle ABC if B = 73.5°, a = 28.2 ft, and c = 46.7 ft. Use the law of cosines to find bbecause we know the lengths of two sides of the triangle and the measure of the included angle.
11. Now use the law of sines to find the measure of another angle. 47.2 ft
12. C b 4 105.2 A B 6 Solve ABC if a = 4, c = 6, and B = 105.2.
13. Solve ABC if b = 9, c = 5, and A = 120.
14. 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.
15. 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.
16. Two boats leave a harbor at the same time, traveling on courses that make an angle of 82°20′ between them. When the slower boat has traveled 62.5 km, the faster one has traveled 79.4 km. At that time, what is the distance between the boats? We are seeking the length of AB.
17. Use the law of cosines because we know the lengths of two sides of the triangle and the measure of the included angle. The two boats are about 94.3 km apart.
18. A lifeguard sits on lifeguard stand that is about eight feet high. He suddenly notices that a swimmer is struggling in the water. The angle between the base of the lifeguard stand and the swimmer is about 102°, and the straight line distance between the base of the stand and the swimmer is about 85 feet. How far is the lifeguard from the swimmer?
19. 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°.
20. 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.
21. 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.
22. Solve triangle ABC if a = 25.4 cm, b = 42.8 cm, and c = 59.3 cm. Use the law of cosines to find the measure of the largest angle, C.
23. Use either the law of sines or the law of cosines to find the measure of angle B. 118.6°
24. 118.6° Alternative:
25. B 8 15 A 11 C Solve ABC if a = 15, b = 11, and c = 8. Solve for A first
26. B 8 15 A 11 C
27. DESIGNING A ROOF TRUSS (SSS) Find the measure of angle B in the figure.
28. Find the measure of angle C in the figure.
29. Four possible cases can occur when solving an oblique triangle.
30. 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
31. 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.)
32. The semiperimeter s is Using Heron’s formula, the area  is
33. The distance “as the crow flies” from Chicago to St. Louis is 262 mi, from St. Louis to New Orleans is 599 mi, and from New Orleans to Chicago is 834 mi. What is the area of the triangular region having these three cities as vertices? The semiperimeter s is
34. The area of the triangular region is about 40,800 mi2.
35. The distance “as the crow flies” from Los Angeles to New York is 2451 mi, from New York to Montreal is 331 mi, and from Montreal to Los Angeles is 2427 mi. What is the area of the triangular region having these three cities as vertices? (Ignore the curvature of Earth.)
36. The semiperimeter is Using Heron’s formula, the area A is