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

2. 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° 20 25 40° 25 20 25 13

3. 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).

4. 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:

5. 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

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

7. The Law of Cosines a b c A B C

8. Solving Oblique Triangles – SAS w/ the Law of Cosines 1. Use the law of cosines to find the side opposite the angle. x 2 = 20 2 + 25 2 – 2(20)(25)cos 40° x  16.1 2. 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.

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