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Law of Cosines

Law of Cosines. Use to find third side of a triangle Or to solve for unknown angles When c is the hypotenuse of a right triangle we get Pythagoras’ theorem!. c 2 = a 2 + b 2 – 2 ab cos ( θ ). Finding a third side length. When you have 2 lengths and the angle between them.

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Law of Cosines

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  1. Law of Cosines • Use to find third side of a triangle • Or to solve for unknown angles • When c is the hypotenuse of a right triangle we get Pythagoras’ theorem! c2 = a2 + b2 – 2abcos(θ)

  2. Finding a third side length • When you have 2 lengths and the angle between them. c2 = a2 + b2 – 2abcosθ = 49 + 144 – 168cos(40°) a = 7 = 64.305 c = 8.02 θ = 40° b = 12

  3. θ = cos-1( ) a2 + b2 – c2 2ab Solving for an angle • When you have all three side lengths c2 = a2 + b2 – 2abcosθ 2abcosθ = a2 + b2 – c2 a = 9 c = 11 θ = 51.75° θ = ? θ = cos-1(0.619) b = 14

  4. Law of Sines ab c • Use to find side length(s) of a triangle • Or to solve for any unknown angle(s) • In a right triangle we get sin(θ) = = = sin(A) sin(B) sin(C) opp hyp

  5. ab sin(A) sin(B) sin(B) = sin(A) b = a· sin(110) sin(40) b = 7· Finding an unknown side length • When you have at least one length and two angles. B = 110° a = 7 b = 10.23 A = 40° b = 10.23 b = ?

  6. Problems • Use the law of cosines to find the measure of the largest angle in a 4-5-6 triangle. • Use the law of sines to find the shortest side in a 40°-60°-80° triangle whose longest side is 10.0 cm. • A triangle has known angles of 37° and 55°. The side between them is 13 cm long. Find the other side lengths. • A plane which is 100 miles due West of you moves in a roughly Northerly direction at 400 mph. If after 10 minutes the new distance to the plane is 130 miles, determine the exact heading of the aircraft.

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