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

W skateparku jeździec podchodzi do zakrzywionej rampy. Jego zmieniająca się stromość zmienia przyspieszenie rowerzysty, podążając po torze ukształtowanym przez ruch kątowy.Gdy rowerzysta porusza się w górę lub w dół rampy, nachylenie ścieżki zmienia się - z podobieństwem do funkcji stycznej, która porównuje wznoszenie pionowe do biegu poziomego.Styczna jest zdefiniowana jako sinus podzielony przez cosinus dla dowolnego kąta mierzonego od poziomu.Z tej definicji można wyprowadzić tożsamość dwukątową. Przypomnij sobie sinus i cosinus o dwukrotnym kącie. Podstawienie ich do definicji stycznej prowadzi do wyniku: tangens dwukrotnie większy od kąta. Kiedy rowerzysta wystrzeliwuje wyżej, a kąt się podwaja, ta tożsamość przewiduje nowe nachylenie.W przypadku mniejszych kątów w pobliżu podstawy rampy, gdzie rowerzysta wchodzi w łagodny zakręt, przydatna staje się tożsamość połówkowego kąta.Przypomnienie sobie i zastosowanie tożsamości półkąta sinusa i cosinusa do definicji stycznej daje wyrażenie. Upraszczając go, biorąc jego sprzęgnięty i mnożąc przez jeden plus lub jeden minus cosinus, otrzymujemy dwie równoważne formy stycznej tożsamości połówkowej.Te tożsamości pomagają w analizie zmian kierunkowych i modelowaniu zakrzywionego ruchu w dynamice skateparku.

tożsamości trygonometryczne ii

Trigonometric Identities II

To&#380;samo&#347;ci trygonometryczne k&#261;ta podw&#243;jnego i k&#261;ta po&#322;&#243;wkowego wywodz&#261; si&#281; ze wzor&#243;w na sum&#281; i r&#243;&#380;nic&#281; k&#261;t&#243;w i s&#322;u&#380;&#261; jako podstawowe narz&#281;dzia do upraszczania wyra&#380;e&#324;, rozwi&#261;zywania r&#243;wna&#324; oraz obliczania ca&#322;ek. To&#380;samo&#347;ci te zmniejszaj&#261; z&#322;o&#380;ono&#347;&#263; funkcji trygonometrycznych, wi&#261;&#380;&#261;c funkcje k&#261;ta wielokrotnego lub u&#322;amkowego z funkcjami k&#261;ta pojedynczego. Ich zastosowania obejmuj&#261; matematyk&#281;, fizyk&#281; i in&#380;ynieri&#281;, szczeg&#243;lnie w analizie Fouriera, mechanice falowej i rachunku r&#243;&#380;niczkowym.To&#380;samo&#347;ci k&#261;ta podw&#243;jnego uzyskuje si&#281; przez podstawienie jednakowych k&#261;t&#243;w do wzor&#243;w na sum&#281; k&#261;t&#243;w. Dla sinusa i cosinusa s&#261; to:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_1_editor_41035.svg' alt='41035_Equation_1.svg' /></figure>Dla tangensa to&#380;samo&#347;&#263; k&#261;ta podw&#243;jnego ma posta&#263;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/41035/41035_Equation_2_editor_41035.svg' alt='41035_Equation_2.svg' /></figure>To&#380;samo&#347;ci te s&#261; szczeg&#243;lnie przydatne do redukcji wy&#380;szych pot&#281;g funkcji trygonometrycznych oraz upraszczania ca&#322;ek.To&#380;samo&#347;ci k&#261;ta po&#322;&#243;wkowego wyprowadza si&#281; ze wzor&#243;w na k&#261;t podw&#243;jny, rozwi&#261;zuj&#261;c wzgl&#281;dem sin^2&#55349;&#57091; i cos^2&#55349;&#57091;. Wyra&#380;a si&#281; je w nast&#281;puj&#261;cy spos&#243;b:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:72.6%;'><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>Wyb&#243;r znaku zale&#380;y od &#263;wiartki, w kt&#243;rej le&#380;y po&#322;owa k&#261;ta. To&#380;samo&#347;ci te s&#261; kluczowe przy wyznaczaniu warto&#347;ci funkcji trygonometrycznych dla k&#261;t&#243;w niestandardowych oraz w metodach ca&#322;kowania, takich jak podstawianie trygonometryczne.

Tożsamości trygonometryczne II

To&#380;samo&#347;ci trygonometryczne k&#261;ta podw&#243;jnego i k&#261;ta po&#322;&#243;wkowego wywodz&#261; si&#281; ze wzor&#243;w na sum&#281; i ...

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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

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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...

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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

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