Law of Cosines c2 = a2 + b2 – 2abcos(θ) Or to solve for unknown angles

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Presentation transcript:

Law of Cosines c2 = a2 + b2 – 2abcos(θ) Or to solve for unknown angles 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!

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

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

Law of Sines a b c = = sin(A) sin(B) sin(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) = opp hyp

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

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.