Name: solving oblique triangles with sines
Uploaded: 2025-11-26T05:44:32.000Z
Duration: 1 min 29 s
Channel: JoVE

Dwie stacje obserwacyjne, oddalone od siebie znaną odległością poziomą, obserwują satelitę pod znanymi kątami elewacji.Odległość od pierwszej stacji do satelity jest znana.Celem jest znalezienie odległości od satelity do drugiej stacji poprzez rozwiązanie trójkąta utworzonego przez dwie stacje i satelitę.Gdy znany jest jeden bok i dwa kąty lub gdy podane są dwa boki i kąt przeciwny do jednego z nich, stosuje się prawo sinusów.Zgodnie z prawem sinusów stosunek długości boku do sinusa jego przeciwnego kąta pozostaje stały w całym trójkącie.W tym przypadku znana jest jedna strona c oraz dwa kąty C i B . Jest to zgodne z prawem sinusów, umożliwiając obliczenie strony b.Weźmy inny przykład. Prawo Sinusów może być użyte do określenia odległości między dwiema flagami na przeciwległych brzegach rzeki.Tutaj bok BC ma 100 metrów, przeciwny kąt A wynosi 30 stopni. Innym znanym kątem jest B o kącie 70 stopni, po przeciwnej stronie nieznanej strony AC.Korzystając z prawa sinusów, BC z kątem A służy do znalezienia AC z kątem B.Ten wynik to pochylona odległość między dwiema flagami wzdłuż bocznego AC.

prawo sinusów

The Law of Sines

Rozwi&#261;zywanie tr&#243;jk&#261;t&#243;w uko&#347;nych, czyli nieprostok&#261;tnych, opiera si&#281; na specyficznych zale&#380;no&#347;ciach trygonometrycznych, przede wszystkim na prawie sinus&#243;w. W odr&#243;&#380;nieniu od tr&#243;jk&#261;t&#243;w prostok&#261;tnych, gdzie zale&#380;no&#347;&#263; d&#322;ugo&#347;ci bok&#243;w opisuje twierdzenie Pitagorasa, tr&#243;jk&#261;ty bez k&#261;ta prostego wymagaj&#261; innego podej&#347;cia. Twierdzenie Pitagorasa ma zastosowanie wy&#322;&#261;cznie w tr&#243;jk&#261;tach prostok&#261;tnych, przez co nie nadaje si&#281; do rozwi&#261;zywania tr&#243;jk&#261;t&#243;w nieprostok&#261;tnych. W takich sytuacjach niezb&#281;dne staje si&#281; prawo sinus&#243;w, kt&#243;re dostarcza zale&#380;no&#347;ci obowi&#261;zuj&#261;cej niezale&#380;nie od warto&#347;ci k&#261;t&#243;w.Prawo sinus&#243;w ustanawia bezpo&#347;redni zwi&#261;zek mi&#281;dzy bokami i k&#261;tami tr&#243;jk&#261;ta, umo&#380;liwiaj&#261;c wyznaczanie nieznanych d&#322;ugo&#347;ci lub k&#261;t&#243;w, o ile dost&#281;pna jest wystarczaj&#261;ca liczba danych. Prawo sinus&#243;w znajduje szerokie zastosowanie w dziedzinach takich jak nawigacja, geodezja i astronomia, gdzie bezpo&#347;redni pomiar wszystkich element&#243;w tr&#243;jk&#261;ta jest cz&#281;sto niewykonalny.Prawo sinus&#243;w g&#322;osi, &#380;e w dowolnym tr&#243;jk&#261;cie ABC stosunki d&#322;ugo&#347;ci bok&#243;w a, b, c, le&#380;&#261;cych naprzeciw odpowiednich k&#261;t&#243;w A, B, C, wzgl&#281;dem sinus&#243;w tych k&#261;t&#243;w s&#261; sta&#322;e. Zale&#380;no&#347;&#263; t&#281; zapisujemy jako:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40077/40077_Equation_1_1__editor_40077.svg' alt='40077_Equation_1_1_.svg' /></figure>Relacja ta obowi&#261;zuje niezale&#380;nie od po&#322;o&#380;enia tr&#243;jk&#261;ta i d&#322;ugo&#347;ci jego bok&#243;w. Szczeg&#243;lnie skutecznie stosuje si&#281; j&#261; w przypadkach, gdy znane s&#261; dwa k&#261;ty i dowolny bok (ASA/AAS), albo gdy znane s&#261; dwa boki i k&#261;t le&#380;&#261;cy naprzeciw jednego z nich (SSA).Prawo sinus&#243;w upraszcza wyznaczanie niewiadomych wielko&#347;ci w tr&#243;jk&#261;tach nieprostok&#261;tnych, usprawniaj&#261;c obliczenia w zastosowaniach praktycznych.

Prawo sinusów

Rozwi&#261;zywanie tr&#243;jk&#261;t&#243;w uko&#347;nych, czyli nieprostok&#261;tnych, opiera si&#281; na specyficznych zale&#380;no&#347;ciach ...

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Math Fundamentals

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Trigonometry

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solving oblique triangles with sines

Solving oblique triangles uses the Law of Sines. An oblique triangle is a triangle with no right angle. These triangles need a different trigonometric method because the Pythagorean theorem works only for right triangles.

The Law of Sines links each side of a triangle to the sine of its opposite angle. In triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The ratios a/sin A, b/sin B, and c/sin C are constant.

This relationship helps find unknown sides or unknown angles when enough information is known. It is especially useful when two angles and any one side are given. It also works when two sides and the angle opposite one of them are known.

The Law of Sines is useful in navigation, surveying, and astronomy. In these fields, direct measurement of every part of a triangle may not be possible. The law makes it easier to solve non-right triangles and find missing measurements.

Solving Oblique Triangles with Sines

radian and degree measure

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Degrees and radians are the two main units used to measure angles. They both describe rotation around a fixed point. A full turn equals 360 degrees or 2π radians, depending on the unit used. Degrees divide a circle into 360 equal parts. This makes them easy to use in everyday math. They are common in navigation, design, and basic geometry. Radians come from the geometry of a circle. One radian is the angle made when the arc length matches the radius of the circle. Since a circle’s...

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Right triangle trigonometry uses sine, cosine, and tangent to connect an acute angle with the sides of the triangle. These trig ratios describe how side lengths relate to one another in right-angled geometry. They give a clear way to measure the links between angles and sides. Sine compares the side opposite an angle to the hypotenuse. Cosine compares the side adjacent to the angle to the hypotenuse. Tangent compares the opposite side to the adjacent side. Each ratio depends on the angle, not...

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Measuring rocket height with trigonometry uses a fixed ground point and a changing angle of elevation. An observer stands a known horizontal distance from the launch site and watches the object rise. The angle between the ground and the line of sight gives the key information for finding height. The tangent function is the main tool for this calculation. Tangent is the ratio of the vertical side to the adjacent side in a right triangle. When the ground distance is fixed, the object’s height...

the law of sines

Solving oblique triangles uses the Law of Sines. An oblique triangle is a triangle with no right angle. These triangles need a different trigonometric method because the Pythagorean theorem works only for right triangles. The Law of Sines links each side of a triangle to the sine of its opposite angle. In triangle ABC, side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. The ratios a/sin A, b/sin B, and c/sin C are constant. This relationship helps find...

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key terms and concepts

Learn how the Law of Sines helps solve oblique triangles by linking sides and angles. Useful for finding missing measurements in non-right triangles.

Solving Oblique Triangles with Sines | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Solving Oblique Triangles with Sines

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