Law of Cosines

MATH 109 - PrecalculusS. Rook

Overview

• Section 6.2 in the textbook:– Law of Cosines

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Law of Cosines

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:

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Abccba cos2222

bc

acbA

2cos

222

Law of Cosines (Continued)

• The Law of Cosines can also be used when finding other side lengths or angles:

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ab

cbaCCabbac

ac

bcaBBaccab

2cos ;cos2

2cos ;cos2

222222

222222

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• Opposite the shortest remaining side

– Guaranteed to be an acute angle– Calculate the measure of the last angle– Always draw the triangle!

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Law of Sines versus Law of Cosines

• When missing the measure of at least two angles in an oblique triangle, the Law of Sines or Law of Cosines must be used:– The Law of Cosines can be used to calculate the measure of

any of the remaining angles• Inverse cosine returns an angle in the interval 0° to 180°

– e.g. cos A = 0.4744 results in A ≈ 61.7°– The Law of Sines can be used to calculate the measure of the

smallest remaining angle• Possible for the larger remaining angle to be obtuse• Inverse sine returns an angle in the interval -90° to 90°

– e.g. sin A = 0.4744 results in A ≈ 28.3° or A ≈ 151.7°

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Law of Sines versus Law of Cosines (Continued)

• No ambiguity if we choose the smallest remaining angle– Smallest remaining angle is guaranteed to be acute

thus there is only one value for the angle– e.g. If we pick the smallest remaining angle we find

that sin A = 0.4744, A ≈ 28.3°• In summary, use the Law of Cosines on any angle or use

the Law of Sines on the angle corresponding to the shortest remaining side

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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

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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!

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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

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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

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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!

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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 6.2

• Next lesson– Linear & Nonlinear Systems of Equations (Section 7.1)

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