Mrs. Valentine's Math and Science - Home



Obj.: I will be able to recognize when to use Law of Sines. I will be able to solve oblique triangles using the Law of Sines. I will be able to apply the Law of Sines to ambiguous cases.VocabularyOblique TrianglesLaw of SinesSAA TriangleASA TriangleAmbiguous Case (SSA)488769512890500NotesLaw of SinesAn ____________ ___________ is a triangle that does _____ contain a ____________ angle. Has _________ __________ angles or _____ ___________ angles and _____ ___________ angle.Relationships for right triangles do not work for oblique triangles.Law of SinesIf A, B, and C are the measures of the angles of a triangle, and a, b, and c are the lengths of the sides opposite these angles, then The _________ of the length of the side of any triangle to the sine of the angle opposite that side is the _________ for all ________ _________ of the triangle.Solving an Oblique Triangle_____________ an _____________ triangle means finding the ______________ of its sides and the ___________________of its angles.Law of Sines can be used to solve _____ and _____ triangles_____ – two ___________ and a _________________ _________ are known._____ – two ___________ and the _______________ _______ are known.Angles can be solved for by remembering the ____________ ___________ ______ ____________ (the three ______________ in a triangle _______ up to _________)To use the Law of Sines to solve for the missing sides, we ________ __________ ______ of the three _____________. The known ratio can be set equal to a second ratio with an unknown side to solve for the side.The _________________ Case (______)In ______, two ____________ and a ________________ _____________ are known. The information given in this case can result in _______, ______, or ____ triangles. In this situation, it is not necessary to draw an accurate sketch. The law of Sines determines the number of triangles, if any, and gives the solution for each triangle.PracticeSolve the following triangles. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.1.2.3.4.Solve the triangles.5.A=57°, B=50°, a=7 6.A=62°, B=54°, a=7 7.A=53°, B=53°, c=4 8.A=40°, B=40°, c=4 Two sides and an angle (SSA) of a triangle are given. Determine whether the measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results.9.a=16, b=19, A=60° 10.a=13, b=23, A=70° Find the variable to the nearest tenth. Do not round until the final answer.11.12.Obj.: I will be able to find the area of an oblique triangle. I will be able to apply the Law of Sines to real-world situations.VocabularyArea of an Oblique TriangleNotesArea of an Oblique TriangleThe _________ of a triangle equals _______________ the _________________ of the lengths of _______ ___________ times the _________ of their included ___________.255951811620500Applications of the Law of SinesSimilar to working with right triangles, the law of sines allows for many different kinds of applied problems. Areas of use include _______________, __________________, ________________, ________________, and the ______________. Can even be used to detect _______________ ____________, like wildfires, through _____________________.PracticeFind the area of the triangle having the given measurements.1.A=47°, b=22ft, c=45ft 2.A=41°, b=25ft, c=37ft Solve each of the following application problems.3.476530838628000The figure shows a 1300 -yard-long sand beach and an oil platform in the ocean. The angle made with the platform from one end of the beach is 80° and from the other end is 75°. Find the distance of the oil platform, to the nearest tenth of a yard, from each end of the beach. 4.467032536830000A tower leans at an angle of about 84.7°. The figure shows that 187 feet from the base of the tower, the angle of elevation to the top is 47.2°. Find the distance, to the nearest foot, from the base to the top of the tower.5.486283092964000The figure shows a shot-put ring. The shot is tossed from A and lands at B. Using modern electronic equipment, the distance of the toss can be measured without the use of measuring tapes. When the shot lands at B an electronic transmitter placed at B sends a signal to a device in the official's booth above the track. The device determines the angles at B and C. At a track meet, the distance from the official's booth to the shot-put ring is 539 feet. If B = 85.1° and C = 5.2°, determine the length of the toss to the nearest tenth of a foot. 6.494030017653000A pier forms an 87° angle with a straight shore. At a distance of 102 feet from the pier, the line of sight to the tip forms a 35° angle. Find the length of the pier to the nearest tenth of a foot. 7.After a wind storm, you notice that your 15 -foot flagpole may be leaning, but you are not sure. From a point on the ground 14 feet from the base of the flagpole, you find that the angle of elevation to the top is 49°. Is the flagpole leaning? If so, find the acute angle, to the nearest degree, that the flagpole makes with the ground. Obj.: I will be able to recognize when to use the Law of Cosines. I will be able to solve an oblique triangle using the Law of Cosines.VocabularyPaceStrideLaw of CosinesSSS TriangleSAS TriangleNotesLaw of Cosines -73088518732500The law of cosines can help ___________________ to study the ________________ of extinct animals, like _______________. Fossilized footprints allow scientists to measure the pace and stride of these creatures._________ – the distance from the left footprint to the next right footprint, and vice versa. __________ – the distance from one left footprint to the next left footprint (or one right footprint to the next)If A, B, and C are the measures of the angles of a triangle, and a, b, and c are the lengths of the sides opposite these angles, then The law of cosines is used to solve _______ and ________ triangles_______ – two _______ and an included _________ are known_______ – all ___________ ___________ are knownSolving Oblique TrianglesSolving an SAS TriangleUse the Law of _____________ to find the side opposite the _________ _________. Use the Law of ___________ to find the angle opposite the _____________ of the two given sides. This angle is always ____________.Find the ____________ _____________ by _________________ the measure of the given angle and the angle found in step 2 from _________.Solving a SSS TriangleUse the Law of _____________ to find the angle opposite the _____________ side.Use the Law of ___________ to find either of the two remaining ___________ angles.Find the ____________ _____________ by _________________ the measure of the given angle and the angle found in step 2 from _________.PracticeSolve the triangles. Round sides to the nearest tenth and angles to the nearest degree. Do not round until the end.1.2.3.4.5. b=9, c=8, A=160° 6.b=24, c=43, A=30° 7.a=10, b=9, c=8 8.a=63 b=23, c=51 910.The three circles are arranged so that they touch each other, as shown in the figures. Use the given radii for the circles with centers A, B, and C, respectively, to solve the triangle.11.5.5, 4.3, 3.3 12.4.9, 3.9, 2.9 Obj.: I will be able to solve applied problems using the Law of Cosine. I will be able to find the area of an oblique triangle using Heron’s Formula.VocabularyHeron’s FormulaNotes____________ FormulaFinds the _______ of a triangle. The __________ of a triangle with sides a, b, and c is where s is one-half its perimeter: ___________________PracticeUse Heron’s formula to find the area of the triangle. Round to the nearest integer as needed.1.a=3ft, b=3ft, c=1ft 2.a=9m, b=14m, c=11m Solve each of the following application problems.3.Use the figure shown to the right to find the pace angle, for the carnivore. Does the angle indicate that this dinosaur was an efficient walker? Describe your answer. 4.Two ships leave a harbor at the same time. One ship travels on a bearing S14°W at 14 miles per hour. The other ship travels on a bearing N75°E at 10 miles per hour. How far apart will the ships be after 3 hours? 5.A plane leaves airport A and travels 590 miles to airport B on a bearing of N35°E. The plane later leaves airport B and travels to airport C 420 miles away on a bearing of S72°E. Find the distance from airport A to airport C to the nearest tenth of a mile. 6.The diagram shows three islands in a bay. You rent a boat and plan to visit each of these remote islands. If you are on island A, on what bearing should you navigate to go to island C? 44533855714007.You are on a fishing boat that leaves its pier and heads east. After traveling for 23 miles, there is a report warning of rough seas directly south. The captain turns the boat and follows a bearing of S35 °W for 14.4 miles. 443928520955000a. At this time, how far are you from the boat's pier??b. What bearing could the boat have originally taken to arrive at this spot? 8.516128037655500The figure shows a 800 foot tower on the side of a hill that forms a 5° angle with the horizontal. Find the length of each of the two guy wires that are anchored 80 feet uphill and downhill from the tower's base and extend to the top of the tower. 9.The dimensions of a triangular lot are 128 feet by 59 feet by 102 feet. If the price of such land is $3 per square foot, how much does the lot cost? ................
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