Law of Cosines

Solving 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? 15 No side opposite from any angle to get the ratio 26° 12.5

Deriving the Law of Cosines Write an equation using Pythagorean theorem for shaded triangle. b h a k c - k A B c

Law of Cosines Similarly Note the pattern

Applying the Cosine Law Now use it to solve the triangle we started with Label sides and angles Side c first C 15 26° 12.5 A B c

Applying the Cosine Law Now calculate the angles use and solve for B C 15 26° 12.5 A B c = 6.65

Applying the Cosine Law The remaining angle determined by subtraction 180 – 93.75 – 26 = 60.25 C 15 26° 12.5 A B c = 6.65 Experiment with Cosine Law Spreadsheet

Wing Span C The leading edge of each wing of the B-2 Stealth Bomber measures 105.6 feet in 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! A

Using the Cosine Law to Find Area Recall that We can use the value for h to determine the area C b h a A B c

Using the Cosine Law to Find Area We can find the area knowing two sides and the included angle Note the pattern C b a A B c

Try It Out Determine the area of these triangles 42.8° 127° 76.3° 17.9 24 12 76.3°

Cost 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. 324 412 516