Math 112 Elementary Functions Section 2 The Law of Cosines Chapter 7 – Applications of Trigonometry

Solving Oblique Triangles – Five Cases Law of Sines (last section) Used to solve AAS, ASA, and SSA triangles. Law of Cosines (this section) Used to solve SAS and SSS triangles. 60° 40° 20 60° 40° 25 60° °

a b c A B C The Law of Cosines (x, y) (c, 0) y x Using the distance formula for side a … Note that this result looks a lot like the Pythagorean theorem. It is when A = 90°. Note: These relationships are also true when A is obtuse (i.e. C is in the second quadrant).

Alternative Proof of the Law of Cosines Case 1: Acute Triangle a b c A B C h xc-x Using the triangle on the left: Using the triangle on the right:

Alternative Proof of the Law of Cosines Case 2: Obtuse Triangle a b c A B C Using the larger right triangle: Using the smaller right triangle: h x

Second Alternative Proof of the Law of Cosines (one more time …) a b c A B C h xy

The Law of Cosines a b c A B C

Solving Oblique Triangles – SAS w/ the Law of Cosines 1. Use the law of cosines to find the side opposite the angle. x 2 = – 2(20)(25)cos 40° x  Use the law of sines to find the smaller of the remaining angles. x/sin 40° = 20/sin  sin  = (20 sin 40°)/x   53.0° 3. Find the third angle.  = 180 – (40° +  )  87.0°  40° 20 x 25  NOTE: Always use EXACT values if possible. Unfortunately, in this case this is not possible. So use as much precision as possible.

Solving Oblique Triangles – SSS w/ the Law of Cosines 1. Use the law of cosines to find one of the angles = – 2(20)(25)cos   = cos -1 [( )/(-2(20)(25)]   28.2° 2. Use the law of cosines to find a second angle = – 2(12)(25)cos   = cos -1 [( )/(-2(12)(25)]   52.0° 3. Find the third angle.  = 180° - (28.2° °) = 99.8°    NOTE: Always use EXACT values if possible.