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

Two observation stations, separated by a known horizontal distance, observe a satellite at known elevation angles.The distance from the first station to the satellite is known.The goal is to find the distance from the satellite to the second station by solving the triangle formed by the two stations and the satellite.When one side and two angles are known, or two sides and an angle opposite to one of them are provided, the Law of Sines is applicable.According to the Law of Sines, the ratio of a side length to the sine of its opposite angle remains constant throughout a triangle.In this case, one side c and two angles C and B are known. This fits the Law of Sines, enabling the calculation of side b.Take another example. The Law of Sines can be used to find the distance between two flags on opposite banks of a river.Here, side BC is 100 meters, opposite angle A of 30 degrees. Another known angle is B of 70 degrees, opposite the unknown side AC.Using the Law of Sines, BC with angle A is used to find AC with angle B.This result is the slanted distance between the two flags along the side AC.

the law of sines

Solving oblique triangles&#8212;those without right angles&#8212;relies on specific trigonometric relationships, most notably the Law of Sines. This is because, unlike right triangles, where the Pythagorean theorem can be used to relate side lengths, oblique triangles lack a 90-degree angle and therefore require a different approach. The Pythagorean theorem is only valid for right-angled triangles, making it unsuitable for solving non-right triangles. In such cases, the Law of Sines becomes essential, as it provides a trigonometric relationship that works regardless of angle size.The Law of Sines provides a direct connection between the sides and angles of a triangle, enabling the determination of unknown lengths or angles when sufficient information is available. Its utility extends across various scientific and technical disciplines, including navigation, surveying, and astronomy, where direct measurement of all elements in a triangle may not be feasible.The Law of Sines asserts that in any triangle ABC, the ratios of the side lengths a, b, and c, which are opposite to angles A, B, and C respectively, to the sine of their corresponding opposite angles are constant. This relationship is mathematically expressed as:<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>This relationship holds true regardless of the triangle's orientation or side lengths. It is especially effective in cases where two angles and any side are known, or when two sides and an angle opposite one of them are provided.The Law of Sines simplifies the process of solving for unknown measurements in non-right triangles, streamlining calculations in practical applications.

The Law of Sines

Solving oblique triangles&#8212;those without right angles&#8212;relies on specific trigonometric relationships, most notably the Law of Sines. This is ...

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Math Fundamentals

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Trigonometry

KEY TERMS AND CONCEPTS

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

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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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