Name: using cosines to find triangle sides
Uploaded: 2025-11-26T05:44:31.000Z
Duration: 1 min 15 s
Channel: JoVE

Consider a triangle with sides a, b, and c, where each angle is named after the vertex opposite its corresponding side.The Law of Cosines states that the square of one side equals the sum of the squares of the other two sides, minus twice the product of those sides multiplied by the cosine of the included angle.It applies when two sides and the included angle are known. It also applies when all three sides are given, and the included angle is found using the inverse of cosine.When the included angle is a right angle, the triangle becomes a right triangle. The cosine of ninety degrees is zero. This removes the cosine term from the formula and simplifies it to the Pythagorean Theorem.In land surveying, the Law of Cosines is used when one side of a marked triangle is inaccessible, such as when it lies across a water body. The other two sides and the included angle are measured using surveying equipment. Their values are then substituted into the Law of Cosines to calculate the length of the unknown side without direct measurement.

the law of cosines

The Law of Cosines is a fundamental result in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It serves as a generalization of the Pythagorean Theorem, enabling calculations in non-right triangles where the simple relationships of right-angled geometry no longer apply. The formula is especially useful in scenarios where direct measurement of one side or angle is not feasible, such as in surveying, navigation, and engineering applications.For any triangle with sides a, b, and c, and corresponding opposite angles A, B, and C, the Law of Cosines is given by:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_1_1__editor_41304.svg' alt='41304_Equation_1_1_.svg' /></figure>Similar expressions can be written for the other sides:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_2_1__editor_41304.svg' alt='41304_Equation_2_1_.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_3_1__editor_41304.svg' alt='41304_Equation_3_1_.svg' /></figure>This law is applicable in two primary situations: when two sides and the included angle are known, or when all three sides are known and the included angle is to be determined.In addition to land surveying, the Law of Cosines is applied in astronomy to calculate distances between celestial bodies forming triangles from Earth's perspective. It is also used in robotics to determine joint angles in arm mechanisms and in physics to resolve forces acting at angles.For example, consider a triangle where the sides a=7 units, b=10 units, and the included angle C=60o. Using the Law of Cosines:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_4_1__editor_41304.svg' alt='41304_Equation_4_1_.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_5_1__editor_41304.svg' alt='41304_Equation_5_1_.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_6_1__editor_41304.svg' alt='41304_Equation_6_1_.svg' /></figure>This shows how the Law of Cosines enables precise distance calculations even in complex geometries.

The Law of Cosines

The Law of Cosines is a fundamental result in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It ...

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using cosines to find triangle sides

The Law of Cosines helps find unknown sides and angles in any triangle. It is a key trig formula for non-right triangles, where the Pythagorean Theorem does not give a direct answer. The law links side lengths to the cosine of an angle, so students can solve triangles with missing measurements.

For a triangle with sides a, b, and c and opposite angles A, B, and C, the formula is written as a squared equals b squared plus c squared minus 2bc cosine A. Similar equations can be used for the other sides. This makes the Law of Cosines useful when two sides and the included angle are known, or when all three sides are known and the missing angle must be found.

The law is often used when direct measurement is hard. Surveyors can use it for land measurements, and it can help in navigation and engineering. It also appears in astronomy to find distances between celestial bodies when they form triangles from Earth’s point of view.

The Law of Cosines also has uses beyond geometry class. Robotics uses it to determine joint angles in arm mechanisms, and physics uses it to resolve forces acting at angles. For example, if a triangle has sides a = 7 units and b = 10 units with an included angle C = 60 degrees, the law can be used to calculate the remaining side. This shows how cosine relationships help solve real triangle problems with precision.

Using Cosines to Find Triangle Sides

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key terms and concepts

Learn how the Law of Cosines helps solve non-right triangles, find missing sides or angles, and supports surveying, navigation, engineering, and more.

Using Cosines to Find Triangle Sides | 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.

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Using Cosines to Find Triangle Sides

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