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

Si consideri un triangolo con i lati a, b e c, in cui ogni angolo prende il nome dal vertice opposto al lato corrispondente.La legge del coseno afferma che il quadrato di un lato è uguale alla somma dei quadrati degli altri due lati, meno il doppio del prodotto di quei lati moltiplicato per il coseno dell'angolo incluso.Si applica quando sono noti due lati e l'angolo incluso. Si applica anche quando tutti e tre i lati sono dati e l'angolo incluso si trova usando l'inverso del coseno.Quando l'angolo incluso è un angolo retto, il triangolo diventa un triangolo rettangolo. Il coseno di novanta gradi è zero. Questo rimuove il termine coseno dalla formula e lo semplifica nel teorema di Pitagora.Nella topografia, la legge del coseno viene utilizzata quando un lato di un triangolo marcato è inaccessibile, ad esempio quando si trova attraverso un corpo idrico. Gli altri due lati e l'angolo incluso vengono misurati utilizzando apparecchiature di rilevamento. I loro valori vengono quindi sostituiti nella legge del coseno per calcolare la lunghezza del lato sconosciuto senza misurazione diretta.

la legge dei coseni

La legge dei coseni &#232; un risultato fondamentale della trigonometria che mette in relazione le lunghezze dei lati di un triangolo con il coseno di uno dei suoi angoli. Costituisce una generalizzazione del teorema di Pitagora, consentendo calcoli in triangoli non rettangoli, dove le semplici relazioni della geometria dei triangoli rettangoli non sono pi&#249; applicabili. La formula &#232; particolarmente utile in situazioni in cui la misurazione diretta di un lato o di un angolo non &#232; possibile, come nelle applicazioni di topografia, navigazione e ingegneria.Per qualsiasi triangolo con lati a, b e c e angoli opposti A, B e C, la legge dei coseni si esprime con la formula:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:21.69%;'><img style='aspect-ratio:830/85;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_1_editor_41304.svg' alt='41304_Equation_1.svg' /></figure>Si ottengono espressioni analoghe per gli altri lati:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:21.56%;'><img style='aspect-ratio:825/85;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_2_editor_41304.svg' alt='41304_Equation_2.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:21.01%;'><img style='aspect-ratio:833/85;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_3_editor_41304.svg' alt='41304_Equation_3.svg' /></figure>Questa legge &#232; applicabile in due situazioni principali: quando sono noti due lati e l'angolo compreso, oppure quando sono noti tutti e tre i lati e si deve determinare l'angolo compreso.Oltre alla topografia, la legge dei coseni viene applicata in astronomia per calcolare le distanze tra i corpi celesti visti dalla prospettiva terrestre. &#200; utilizzata anche in robotica per determinare gli angoli di giunto nei bracci robotici e in fisica per scomporre forze agenti con un angolo.Ad esempio, consideriamo un triangolo in cui i lati a = 7 unit&#224;, b = 10 unit&#224; e l'angolo compreso c = 60&#176;. Utilizzando la legge dei coseni:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:25%;'><img style='aspect-ratio:1083/85;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_4_editor_41304.svg' alt='41304_Equation_4.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:19.78%;'><img style='aspect-ratio:928/74;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_5_editor_41304.svg' alt='41304_Equation_5.svg' /></figure><figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:17.04%;'><img style='aspect-ratio:721/80;' src='https://cloudfront.jove.com/files/ftp_upload/41304/41304_Equation_6_editor_41304.svg' alt='41304_Equation_6.svg' /></figure>Questo esempio mostra come la legge dei coseni consenta calcoli precisi delle distanze anche in geometrie complesse.

La legge dei coseni

La legge dei coseni &#232; un risultato fondamentale della trigonometria che mette in relazione le lunghezze dei lati di un triangolo con il coseno di uno ...

Education

Didattica

JoVE Core

Math Fundamentals

Fondamenti di Matematica

Unit

Unità

Trigonometry

Trigonometria

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

The Law of Cosines

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

radian and degree measure

Degrees to Radians in Circle Measure

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...

trigonometric functions of real numbers

Unit Circle Trig: Signs and Symmetry

The unit circle is the basis for trigonometric functions on real numbers. It is a circle with radius one centered at the origin of the coordinate plane. On this circle, arc length means the distance measured along the circumference between two points. Each real number t is shown as an arc length measured counterclockwise from the positive x-axis. A point on the unit circle has coordinates (x, y). These coordinates define cosine and sine. The x-coordinate equals cos t, and the y-coordinate...

graphs of trigonometric functions

Trigonometric Wave Graphs and Asymptotes

Trigonometric functions create repeating waves and curved graphs with clear symmetry. Their shapes come from the unit circle, where a point moves counterclockwise and its vertical and horizontal positions form the sine and cosine values. Sine and cosine graphs are smooth waves with the same period, or repeat length. Their ranges stay bounded between -1 and 1. Sine, cosine, and the other trig functions all show regular patterns, but each one has its own graph shape. Tangent is the ratio of...

trigonometry of right triangles

Trig Ratios for Right Triangle Sides

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...

trigonometric functions: problem solving

Measuring Rocket Height with Trigonometry

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 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...

the law of cosines

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...

trigonometric identities i

Unit Circle Trig Identities and Sum Formulas

Trigonometric identities link trigonometric functions and stay true for every angle in their domains. A key example is the Pythagorean identity, which comes directly from the geometry of the unit circle. For an angle θ, a point on the unit circle has coordinates (cos θ, sin θ). Because the radius is 1, the Pythagorean Theorem gives the basic identity. That same identity can be rearranged to create another useful form. If it is divided by cos²θ, assuming cos θ is not 0, it leads to a second...

trigonometric identities ii

Double-Angle and Half-Angle Formulas

Double-angle and half-angle trigonometric formulas connect angles that are multiplied or divided by two. They come from the sum and difference formulas, so they build on earlier trigonometry skills. These identities help simplify expressions, solve equations, and evaluate integrals. The double-angle formulas come from substituting equal angles into the sum formulas. For sine and cosine, they link the value at 2θ to the value at θ. The tangent double-angle identity follows the same idea. These...

trigonometric identities iii

Cofunction Identities and Complementary Angles

Cofunction identities show how trigonometric functions change for complementary angles. Complementary angles add up to 90°. These identities are a useful part of trigonometry for grades 9–12. The idea becomes clear on the unit circle. Every angle θ, measured counterclockwise from the positive x-axis, matches a point with coordinates (cos θ, sin θ). The cosine gives the horizontal component, and the sine gives the vertical component of the terminal side. The same point can also be described...

inverse trigonometric functions

Inverse Trig Functions and Angle Finding

Inverse trigonometric functions are used to find angles from trig ratios. Standard trigonometric functions turn angles into ratios, while inverse trigonometric functions reverse that process and map a ratio back to its angle. That makes them useful whenever a problem gives distances or side lengths instead of an angle. A function must be one-to-one to have a valid inverse. This means each input must match only one output. Trigonometric functions are not naturally one-to-one over their full...

trigonometric equations

Solving Trigonometric Equations

Trigonometric equations use one or more trigonometric functions, and they often appear in mathematical modeling. Some are identities, which are true for every value of the variable. Others are conditional equations, which are true only for specific values. Solving these equations usually combines algebra with the basic properties of trigonometric functions. Some equations look like standard algebra problems, so factoring can work well. For example, a quadratic form in sin x can be factored...

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.

math fundamentals

Using Cosines to Find Triangle Sides

Description

Transcript

Tags