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Law of Cosines

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. B. In any triangleABC: a 2 = b 2 + c 2 -2bc(cos A)

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Law of Cosines

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  1. 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

  2. The Law of Cosines B • In any triangleABC: • a2 = b2 + c2 -2bc(cos A) • b2 = a2 + c2 -2ac(cosB) • c2 = a2 + b2 - 2ab(cosC) a c C b A

  3. Find c to the nearest tenth using Law of Cosines • c2 = a2 + b2 -2ab(cosV) • c2 = 802 + 602 -2(80)(60)(cos 22) • c2 = 1099.03 • c = 33.2 T 60 c 22o U V 80

  4. Finding missing angle with Law of Cosines • Find m<C to nearest degree • c2 = a2 + b2 -2ab(cos C) • 622 = 322 + 572 - 2(32)(57)(cosC) • -429 = -3648(cosC) • cosC = .118 • cos-1(.118 )= 83.22 • <C = 83.22 C 32 57 62 D E

  5. Find < Y to the nearest tenth • y2 = x2 + z2 -2xz(cosY) • 502 = 312 + 252 -2(31)(25)(cosY) • 914= -1550(cosY) • cosY= -.59 • cos-1(-.59) = 126.16 Y 25 X 31 50 Z

  6. 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

  7. Using Law of Cosines with a right triangle • Find y to the nearest tenth. • y2 = r2 + t2 -2rt(cosS0 • y2 = 122 + 82 -2(12)(8)(cos90) • y2 = 122 + 82 - 0 Pythagorean theorem? • y2 = 144 + 64 • y = 14.4 R y 8 8 T S 12

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

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

  10. Heron’s Formula

  11. Using Heron’ formula

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