MA 15400 Lesson 19 Section 8.2Solve for ABCAngle of a box The rectangular box shown in the figure has dimensions 2' X 6' X 8'. Approximate the angle formed by a diagonal of the base and a diagonal of the 2' X 8' side.MA 15400 Lesson 19 Section 8.2The Law of CosinesSolve for ABC = 60, b = 20, c = 30A 60.0°BCab 20.0c 30.0 = 7350', c = 14.0, a = 87.0 AB 73.8°Ca 87.0bc 14.01The Law of Cosines (General Form):The square of the length of any side of a triangle equals the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of the other two sides and the cosine of the angle between them.The Law of Cosines:If ABC is a triangle labeled in the usual manner, then,(1) a2b2 c2 2bc cos(2) b2a2 c2 2ac cos(3) c2a2 b2 2ab cosUsually we never use the law of cosines more than once when solving a triangle. To find remaining parts, we usually use the law of sines.When using the law of sines, always try to find the angle opposite the smallest side first. It is guaranteed to be an acute angle. If you try to find an angle opposite the largest side, the angle could be obtuse.Always draw a triangle!MA 15400 Lesson 19 Section 8.2The Law of Cosinesa = 6.0, b = 3.0, c = 4.0ABCa 6.0b 3.0c 4.0Distance between ships. At 2:00 PM, a ship leaves port and travels N40E at the rate of 30 mph. Another ship leaves the same port at 3:00 PM, and travels N75W at the rate of 20 mph. Approximately how far apart are the ships at 4:30 PM? Flight distance An airplane flies 200 miles from point A in the direction 40 and then travels in the2We can now do navigation problems that donot necessarily lead to right triangles.MA 15400 Lesson 19 Section 8.2The Law of Cosinesdirection 100 for 300 miles. a) Approximately how far is the airplane from A?b) What direction is needed for the plane to return back to point A?Surveying: To find the distance from the house at A to the house at B, a surveyor measures the angle BAC to be 40 and then walks off a distance of 100 feet to C and measures the angle ACB to be 60. What is the distance from A to B?3A40°100°Angle fordirection?°MA 15400 Lesson 19 Section 8.2The Law of CosinesLandscaping Pat needs to determine the height of a tree before cutting it down to the sure that itwill not fall on a nearby fence. The angle of elevation of the tree from one position on a flat path from the tree is 30, and from a second position 40 feet father along this path it is 20. What is the height of the tree?Angle of a box The rectangular box shown in the figure has dimensions 2' X 6' X 8'. Approximate the angle formed by a diagonal of the base and a diagonal of the 2' X 8' side.4BARiverC4060100 ft.2'4'8'h20°30°40We need a 3rd side to form a triangle. Draw that side. Label the 3 sides x,y, and z.It is easy to find the lengths of the 3sides; they are diagonals of rectangles.MA 15400 Lesson 19 Section 8.2The Law of Cosines
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