Law of Cosines Like the Law of Sines

Law of Cosines Like the Law of Sines, the Law of Cosines relates the sides and angles of any triangle. The Law of Cosines can be used when there is not enough information to use the Law of Sines

The Law of Cosines In any triangle. ABC: B a c a 2 = b 2 + c 2 -2 bc(cos A) C b b 2 = a 2 + c 2 -2 ac(cos. B) c 2 = a 2 + b 2 - 2 ab(cos. C) A

Find c to the nearest tenth using Law of Cosines c 2 = a 2 + b 2 -2 ab(cos. V) c 2 = 802 + 602 -2(80)(60)(cos 22) c 2 = 1099. 03 c = 33. 2 T c 60 22 o U 80 V

Finding missing angle with Law of Cosines Find m<C to nearest degree c 2 = a 2 + b 2 -2 ab(cos C) 622 = 322 + 572 - 2(32)(57)(cos. C) -429 = -3648(cos. C) cos. C =. 118 57 cos-1(. 118 )= 83. 22 E <C = 83. 22 C 32 62 D

Find < Y to the nearest tenth y 2 = x 2 + z 2 -2 xz(cos. Y) 502 = 312 + 252 -2(31)(25)(cos. Y) 914= -1550(cos. Y) cos. Y= -. 59 25 X -1 cos (-. 59) = 126. 16 Y 31 50 Z

Hint Remember trig ratios refer to right triangles only, but the Law of Sines and the Law of Cosines can be used in any triangles

Using Law of Cosines with a right triangle Find y to the nearest tenth. R y y 2 = r 2 + t 2 -2 rt(cos. S 0 T y 2 = 122 + 82 -2(12)(8)(cos 90) 12 y 2 = 122 + 82 - 0 Pythagorean theorem? y 2 = 144 + 64 y = 14. 4 8 8 S

practice 1. find c to the nearest whole number P 92 o 48 40 N c 2. find m<X to the nearest degree X 14 12 Y 7 Z M

practice Use Law of Cosines to find a to the nearest tenth. M a L 14 27 N

Heron’s Formula

Using Heron’ formula