Name: cofunction identities and complementary angles
Uploaded: 2025-11-26T05:44:26.000Z
Duration: 1 min 27 s
Channel: JoVE

Tożsamości współfunkcyjne opisują, w jaki sposób pary funkcji trygonometrycznych są ze sobą powiązane, gdy ich kąty sumują się do 90 stopni.Dwie podstawowe tożsamości współfunkcyjne można udowodnić, biorąc pod uwagę punkt na okręgu jednostkowym.Niech theta będzie kątem z dodatnią osią x. Z definicji współrzędne x i y są wtedy odpowiednio cosinus theta i sinus theta.Dla tego samego punktu zmierz jego kąt od dodatniej osi y, 90 stopni minus theta.Współrzędne punktu są równe sin 90 minus theta i cos 90 minus theta.Ponieważ współrzędne pozostają niezmienione, zrównanie wyrażeń dowodzi tożsamości współfunkcyjnych dla sinusa i cosinusa.Jednak wszystkie inne funkcje trygonometryczne, które są oparte na sinusie i cosinusie, dziedziczą tę właściwość.Te tożsamości pomagają w obliczeniach w świecie rzeczywistym.Na przykład, gdy łódź znajduje się daleko od latarni morskiej, odległość jest mierzona za pomocą kąta wzniesienia od poruszającej się łodzi do szczytu latarni morskiej. Tożsamości współfunkcyjne pozwalają na zastąpienie tangensa theta tangensem kąta komplementarnego we wzorze odległości, zachowując wynik z elastycznością w obliczeniach.

tożsamości trygonometryczne iii

Trigonometric Identities III

To&#380;samo&#347;ci kofunkcyjne s&#261; kluczowym zagadnieniem w trygonometrii. Opisuj&#261; relacje mi&#281;dzy funkcjami trygonometrycznymi, gdy ich k&#261;ty s&#261; dope&#322;niaj&#261;ce si&#281; &#8212; to znaczy ich suma wynosi 90&#176;. Na okr&#281;gu jednostkowym ka&#380;dy k&#261;t &#952;, mierzony przeciwnie do ruchu wskaz&#243;wek zegara od dodatniej osi x, odpowiada punktowi o wsp&#243;&#322;rz&#281;dnych (cos &#952;, sin &#952;). Warto&#347;ci te reprezentuj&#261; odpowiednio poziom&#261; i pionow&#261; sk&#322;adow&#261; ramienia ko&#324;cowego k&#261;ta.Je&#347;li ten sam punkt na okr&#281;gu jednostkowym opiszemy k&#261;tem dope&#322;niaj&#261;cym 90&#176; &#8722; &#952;, punktem odniesienia staje si&#281; dodatnia o&#347; y. W&#243;wczas odcinek poziomy, pierwotnie przyleg&#322;y do k&#261;ta &#952;, staje si&#281; bokiem przeciwnym dla k&#261;ta 90&#176; &#8722; &#952;, natomiast odcinek pionowy, kt&#243;ry wcze&#347;niej by&#322; przeciwny, staje si&#281; przyleg&#322;y. Zatem wsp&#243;&#322;rz&#281;dne mo&#380;na zapisa&#263; r&#243;wnowa&#380;nie jako (sin (90&#176; &#8722; &#952;), cos (90&#176; &#8722; &#952;)). Skoro oba opisy dotycz&#261; tego samego punktu, wynika st&#261;d, &#380;e:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:40.32%;'><img style='aspect-ratio:1425/74;' src='https://cloudfront.jove.com/files/ftp_upload/41457/41457_Equation_1_editor_41457.svg' alt='41457_Equation_1.svg' /></figure>Te r&#243;wno&#347;ci to podstawowe to&#380;samo&#347;ci kofunkcyjne, ukazuj&#261;ce zamienno&#347;&#263; r&#243;l sinusa i cosinusa dla k&#261;t&#243;w dope&#322;niaj&#261;cych si&#281;.Opr&#243;cz sinusa i cosinusa analogiczne rozumowanie dotyczy pozosta&#322;ych funkcji trygonometrycznych, co daje pe&#322;ny zestaw to&#380;samo&#347;ci kofunkcyjnych:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:40.71%;'><img style='aspect-ratio:1444/361;' src='https://cloudfront.jove.com/files/ftp_upload/41457/41457_Equation_2_editor_41457.svg' alt='41457_Equation_2.svg' /></figure>To&#380;samo&#347;ci te demonstruj&#261; symetri&#281; funkcji trygonometrycznych i ich &#347;cis&#322;y zwi&#261;zek z k&#261;tami dope&#322;niaj&#261;cymi w tr&#243;jk&#261;tach prostok&#261;tnych oraz na okr&#281;gu jednostkowym. Zapewniaj&#261; systematyczny spos&#243;b przechodzenia mi&#281;dzy funkcjami podczas pracy z k&#261;tami mierzonymi wzgl&#281;dem r&#243;&#380;nych kierunk&#243;w odniesienia.

Tożsamości trygonometryczne III

To&#380;samo&#347;ci kofunkcyjne s&#261; kluczowym zagadnieniem w trygonometrii. Opisuj&#261; relacje mi&#281;dzy funkcjami trygonometrycznymi, gdy ich ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Trigonometry

Trygonometria

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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 using the complementary angle 90° − θ. In that form, the reference changes to the positive y-axis. The horizontal segment that was adjacent to θ becomes opposite to 90° − θ, and the vertical segment that was opposite becomes adjacent.

Because both descriptions refer to the same point, the coordinates can be written as (sin(90° − θ), cos(90° − θ)). This leads to the basic cofunction identities, where sine and cosine switch roles for complementary angles. The same pattern also applies to the other trigonometric functions, giving the full set of cofunction identities.

These identities show the symmetry of trigonometric functions. They also connect right-triangle relationships with the unit circle. That makes them a systematic way to change from one trig function to another when angles are measured from different reference directions.

Cofunction Identities and Complementary Angles

cofunction-identities-and-complementary-angles

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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...

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key terms and concepts

Learn how cofunction identities connect sine, cosine, and other trig functions for complementary angles on the unit circle and in right triangles.

Cofunction Identities and Complementary Angles | 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

Cofunction Identities and Complementary Angles

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