Name: double-angle and half-angle formulas
Uploaded: 2025-11-26T05:44:27.000Z
Duration: 1 min 28 s
Channel: JoVE

In uno skate park, un rider si avvicina a una rampa curva. La sua pendenza mutevole altera l'accelerazione del ciclista, tracciando un percorso modellato dal movimento angolare.Man mano che il ciclista sale o scende dalla rampa, la pendenza del percorso cambia, con somiglianze con la funzione tangente, che confronta l'ascesa verticale con la corsa orizzontale.La tangente è definita come seno diviso per il coseno per ogni angolo misurato dall'orizzontale.Da questa definizione si può derivare l'identità a doppio angolo. Ricorda seno e coseno del doppio dell'angolo. Sostituendo questi nella definizione di tangente si ottiene il risultato: tangente del doppio dell'angolo. Quando il ciclista si lancia più in alto e l'angolo raddoppia, questa identità predice la nuova pendenza.Per angoli più piccoli vicino alla base della rampa, dove il ciclista entra in una curva dolce, l'identità del semiangolo diventa utile.Ricordando e applicando le identità del semiangolo seno e coseno alla definizione della tangente si ottiene un'espressione. Semplificandola prendendo il suo coniugato e moltiplicandolo per uno più o uno meno coseno si ottengono due forme equivalenti dell'identità semiangolo tangente.Queste identità aiutano ad analizzare i cambiamenti di direzione e a modellare il movimento curvo nelle dinamiche degli skate park.

identità trigonometriche ii

Trigonometric Identities II

Le identit&#224; trigonometriche a doppio angolo e a semiangolo derivano dalle formule fondamentali di somma e differenza e servono come strumenti essenziali per semplificare espressioni, risolvere equazioni e valutare integrali. Queste identit&#224; riducono la complessit&#224; delle funzioni trigonometriche mettendo in relazione funzioni di un angolo multiplo o di una frazione di angolo con funzioni di un singolo angolo. Le loro applicazioni si estendono alla matematica, alla fisica e all&#8217;ingegneria, in particolare nell&#8217;analisi di Fourier, nella meccanica ondulatoria e nel calcolo infinitesimale.Le identit&#224; a doppio angolo si ottengono sostituendo lo stesso angolo nelle formule di somma. Per seno e coseno, valgono:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:49.03%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_1_editor_41035.svg' alt='41035_Equation_1.svg' /></figure>Per la tangente, l&#8217;identit&#224; a doppio angolo &#232;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:49.07%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_2_editor_41035.svg' alt='41035_Equation_2.svg' /></figure>Queste identit&#224; sono particolarmente utili per ridurre potenze elevate di funzioni trigonometriche e semplificare integrali.Le identit&#224; a semiangolo si ricavano dalle formule di doppio angolo risolvendo rispetto a sin^2 &#952; e cos^2 &#952;. Si esprimono come:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:70.06%;'><img style='aspect-ratio:1920/650;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_3_editor_41035.svg' alt='41035_Equation_3.svg' /></figure>La scelta del segno dipende dal quadrante in cui ricade il semiangolo. Queste identit&#224; sono cruciali nella valutazione delle funzioni trigonometriche per angoli non notevoli e nelle tecniche di integrazione, come la sostituzione trigonometrica.

Identità Trigonometriche II

Le identit&#224; trigonometriche a doppio angolo e a semiangolo derivano dalle formule fondamentali di somma e differenza e servono come strumenti ...

Education

Didattica

JoVE Core

Math Fundamentals

Fondamenti di Matematica

Unit

Unità

Trigonometry

Trigonometria

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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 formulas are useful when simplifying higher powers of trigonometric expressions.

The half-angle formulas are found by solving the double-angle equations for sin 2θ and cos 2θ. They give trigonometric values for half of an angle instead of the whole angle. The sign in a half-angle formula depends on the quadrant where the half-angle lies.

These identities are important for finding trigonometric values for nonstandard angles. They also appear in calculus, especially in trigonometric substitution and in methods for evaluating integrals. Their uses also extend into mathematics, physics, and engineering, including Fourier analysis and wave mechanics.

Double-Angle and Half-Angle Formulas

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

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...

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 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 double-angle and half-angle trig formulas come from sum and difference identities, and how they help simplify equations, powers, and integrals.

Double-Angle and Half-Angle Formulas | 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

Double-Angle and Half-Angle Formulas

Description

Transcript

Tags