1. Law of Cosines 2014 Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 An oblique triangle is a triangle that has no right angles. Definition: Oblique Triangles To solve an oblique triangle, you need to know the measure of at least one side and the measures of any other two parts of the triangle – two sides, two angles, or one angle and one side. C BA a b c

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 The following cases are considered when solving oblique triangles. Solving Oblique Triangles 1.Two angles and any side (AAS or ASA) 2. Two sides and an angle opposite one of them (SSA) 3. Three sides (SSS) 4. Two sides and their included angle (SAS) A C c A B c a c b C c a c a B

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 The last two cases (SSS and SAS) can be solved using the Law of Cosines. (The first two cases can be solved using the Law of Sines.) Definition: Law of Cosines Law of Cosines Standard FormAlternative Form

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Find the three angles of the triangle. Example: Law of Cosines - SSS Example: C BA 8 6 12 Find the angle opposite the longest side first. Law of Sines: 36.3  117.3  26.4 

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Solve the triangle. Example: Law of Cosines - SAS Example: 67.8  Law of Sines: 37.2  C BA 6.2 75  9.5 9.9 Law of Cosines:

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Definition: Heron’s Area Formula Example: Find the area of the triangle. 5 10 8

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Definition: Heron’s Area Formula Heron’s Area Formula Given any triangle with sides of lengths a, b, and c, the area of the triangle is given by Example: Find the area of the triangle. 5 10 8

9. You Try: The pitcher’s mound on a women’s softball field is 43 feet from home plate and the distance between bases is 60 feet, as shown. (The pitcher’s mound is not halfway between home plate and second base) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 How far is the pitcher’s mound from first base? 42.43 ft.

10. A ship travels 60 miles due east, then adjusts its course northward as shown in the figure. After traveling 80 miles in that direction, the ship is 139 miles from its point of departure. Describe the bearing from point B to point C. N 76.15 0

11. A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 pm the ship changes course to N 54 0 W. Find the ship’s bearing and distance from the port of departure at 3 pm. N a b c 54 0 C A 144 0 N 57.4 nm 57.4 sin 144 0 20 11.82 0 B 78.18 0

12. A plane flies 810 miles from Niagara to Cuyahoga with a bearing of 75 0. Then it flies 648 miles from Cuyahoga to Rosemount with a bearing of 32 0. Draw a figure to visually represent the problem, and then find the straight-line distance and bearing from Niagara to Rosemount. N N 810 mi C N 648 mi R 75 0 32 0 c r n 75 0 137 0 15 0

13. Homework Pg. 405 1-9 odd, 19-31 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13