Law of CosinesWho's Law Is It, Anyway?Solving an SAS TriangleDeriving the Law of CosinesSlide 5Applying the Cosine LawSlide 7Slide 8Wing SpanAssignment AUsing the Cosine Law to Find AreaSlide 12Try It OutCost of a LotAssignment BLaw of CosinesLesson 4.22Who's Law Is It, Anyway?Murphy's Law:Anything that can possibly go wrong, will go wrong (at the worst possible moment).Cole's Law ??Finely chopped cabbage3Solving an SAS TriangleThe Law of Sines was good forASA - two angles and the included side AAS - two angles and any sideSSA - two sides and an opposite angle (being aware of possible ambiguity)Why would the Law of Sines not work for an SAS triangle? 1512.526°No side opposite from any angle to get the ratioNo side opposite from any angle to get the ratio4Deriving the Law of CosinesWrite an equationusing Pythagorean theorem for shaded triangle. b h ak c - kA BCcsincosh b Ak b A= �= �( ) ( )( )2 222 2 2 2 2 22 2 2 2 22 2 2sin cossin 2 cos cossin cos 2 cos2 cosa b A c b Aa b A c c b A b Aa b A A c c b Aa b c c b A= � + - �= + - ��� += + + - ���= + - ���5Law of CosinesSimilarlyNote the pattern2 2 22 2 22 2 22 cos2 cos2 cosa b c c b Ab a c a c Bc b a a b C= + - ���= + - ���= + - ���6Applying the Cosine LawNow use it to solve the triangle we started withLabel sidesand anglesSide c first1512.526°A BCc2 2 22 22 cos15 12.5 2 15 12.5 cos 26c b a a b Cc= + - ���= + - � � �o7Applying the Cosine LawNow calculate the anglesuseand solve for B1512.526°A BCc = 6.652 2 22 cosb a c a c B= + - ���2 2 2 2 2 21cos cos2 2b a c b a cB Ba c a c-� �- - - -= =� �- �� - ��� �8Applying the Cosine LawThe remaining angledetermined by subtraction180 – 93.75 – 26 = 60.251512.526°A BCc = 6.65Experiment with Cosine Law SpreadsheetExperiment with Cosine Law Spreadsheet9Wing SpanThe leading edge ofeach wing of theB-2 Stealth Bombermeasures 105.6 feetin length. The angle between the wing's leading edges is 109.05°. What is the wing span (the distance from A to C)?Hint … use the law of cosines!AC10Assignment ALesson 4.2APage 308Exercises 1 – 27 odd, and 41 - 51 odd11Using the Cosine Law to Find AreaRecall thatWe can use the value for hto determinethe area b h aA Bcsinh b A= �1sin2Area c b A= ��C12Using the Cosine Law to Find AreaWe can find the area knowing two sides and the included angleNote the pattern 1sin21sin2Area a b Cc a B= ��= �� b aA BcC13Try It OutDetermine the area of these triangles127°122476.3°42.8°17.914Cost of a LotAn industrial piece of real estate is priced at $4.15 per square foot. Find, to the nearest $1000, the cost of a triangular lot measuring 324 feet by 516 feet by 412 feet.51641232415Assignment BLesson 4.2BPage 309Exercises 29 – 39 oddand 53 – 61