Presentation on theme: "Law of Cosines c 2 = a 2 + b 2 – 2abcos(θ) Use to find third side of a triangle Or to solve for unknown angles When c is the hypotenuse of a right triangle."— Presentation transcript:
Law of Cosines c 2 = a 2 + b 2 – 2abcos(θ) 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. a = 7 b = 12 θ = 40° c 2 = a 2 + b 2 – 2abcosθ = 49 + 144 – 168cos(40°) = 64.305 c = 8.02
Solving for an angle When you have all three side lengths a = 9 b = 14 θ = ? c = 11 c 2 = a 2 + b 2 – 2abcosθ 2abcosθ = a 2 + b 2 – c 2 θ = cos -1 ( ) a 2 + b 2 – c 2 2ab θ = cos -1 (0.619) θ = 51.75°
Law of Sines 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) = a b c sin(A) sin(B) sin(C) == opp hyp
Finding an unknown side length When you have at least one length and two angles. a = 7 b = ? A = 40° B = 110° sin(A) sin(B) a b = sin(A) sin(B) b = a· sin(40) sin(110) b = 7· b = 10.23
Problems 1.Use the law of cosines to find the measure of the largest angle in a 4-5-6 triangle. 2.Use the law of sines to find the shortest side in a 40°- 60°-80° triangle whose longest side is 10.0 cm. 3.A triangle has known angles of 37° and 55°. The side between them is 13 cm long. Find the other side lengths. 4.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.