Law of Cosines Trigonometry MATH 103 S. Rook

Overview Section 7.3 in the textbook: – Law of Cosines: SAS case – Law of Cosines: SSS case 2

Law of Cosines: SAS Case

Law of Cosines Recall the four cases of we discussed in the last lesson: – AAS/ASA, SSA, SAS, SSS – The first two are handled by the Law of Sines The last two cases are handled by the Law of Cosines: – When we are looking for the length of side A – Can be algebraically manipulated when looking for the measure of angle A: 4

Law of Cosines (Continued) The Law of Cosines can also be used when finding other side lengths or angles: 5

Law of Cosines – SAS Some strategies when solving a triangle in the SAS case (know two sides and the angle opposite the third side): – Use the Law of Cosines to calculate the length of the missing side – Use either the Law of Cosines to calculate the measure of either remaining angle OR use the Law of Sines to calculate the measure of the smallest remaining angle Smallest angle is opposite the shortest remaining side – Guaranteed to be an acute angle – Calculate the measure of the last angle – Always draw the triangle! 6

Law of Cosines – SAS (Example) Ex 1: Draw the triangle and solve for the remaining components: a)B = 10° 35’, a = 40, c = 9 b)A = 71°, b = 5, c = 10 7

Law of Cosines: SSS Case

Law of Cosines – SSS Some strategies when solving a triangle in the SSS (all three sides) case: – Use the Law of Cosines to solve for the measure of the angle opposite the longest side Each triangle has at most one obtuse angle (why?) and it is guaranteed to be opposite the longest side – Use the Law of Sines or Law of Cosines to find the measure of any of the remaining angles Both remaining angles are guaranteed to be acute thus there is no possibility for ambiguity – Calculate the measure of the last angle – Always draw the triangle! 9

Law of Cosines – SSS (Example) Ex 2: Draw the triangle and solve for the remaining components: a)a = 11, b = 17, c = 20 b)a = 50, b = 100, c = 75 10

Law of Cosines – Application (Example) Ex 3: A boat race runs along a triangular course marked by buoys A, B, and C. The race starts at buoy A with the boats headed due west for 3700 meters where they reach buoy B. The boats then turn northeast and travel 1700 meters until they reach buoy C. The final leg of the race takes the boats 3000 miles back to buoy A. Find the bearing of a) buoy C from buoy B b) buoy A from buoy C 11

Final Notes on Oblique Triangles You must learn to become proficient in determining which of the 4 cases (AAS/ASA, SSA, SAS, SSS) applies to a particular oblique triangle – i.e. You will not always be told when to use either the Law of Sines or Law of Cosines! 12

Summary After studying these slides, you should be able to: – Apply the Law of Cosines in solving for the components of a triangle or in an application problem Additional Practice – See the list of suggested problems for 7.3 Next lesson – Study for the Final Exam! 13