Name: unit circle trig identities and sum formulas
Uploaded: 2025-11-26T05:44:29.000Z
Duration: 1 min 27 s
Channel: JoVE

Tożsamość trygonometryczna to równanie obejmujące funkcje trygonometryczne, które jest prawdziwe dla wszystkich kątów.Jedna z takich tożsamości, zwana tożsamością pitagorejską, może być wyprowadzona za pomocą okręgu jednostkowego.Każdy punkt na tym okręgu odpowiada kątowi mierzonemu od dodatniej osi x.Współrzędna x punktu jest cosinusem kąta, a współrzędna y jest sinusem kąta.Współrzędne x i y tworzą prostopadłe boki trójkąta prostokątnego, z promieniem jako przeciwprostokątną.Zgodnie z twierdzeniem Pitagorasa kwadrat współrzędnej y plus kwadrat współrzędnej x jest równy kwadratowi promienia.Ponieważ promień jest równy jeden, dowodzi to, że sinus do kwadratu plus cosinus do kwadratu równa się jeden.Jest to znane jako tożsamość pitagorejska i odnosi się do wszystkich rzeczywistych punktów widzenia.Ta tożsamość, wyprowadzona w dwóch wymiarach, ma zastosowania, które rozciągają się na systemy trójwymiarowe. W astronomii współrzędne niebieskie opisują położenie gwiazd na sferze niebieskiej.Identyczność zapewnia, że te pozycje pozostają spójne na sferze, co pozwala astronomom śledzić gwiazdy, które wydają się poruszać po niebie.

tożsamości trygonometryczne i

To&#380;samo&#347;ci trygonometryczne to r&#243;wnania wi&#261;&#380;&#261;ce funkcje trygonometryczne, kt&#243;re s&#261; spe&#322;nione dla wszystkich k&#261;t&#243;w w ich dziedzinach. Podstawow&#261; z nich jest to&#380;samo&#347;&#263; pitagorejska, wynikaj&#261;ca bezpo&#347;rednio z geometrii okr&#281;gu jednostkowego. Dla dowolnego k&#261;ta &#952;, punkt na okr&#281;gu jednostkowym ma wsp&#243;&#322;rz&#281;dne (cos &#952;, sin &#952;), a poniewa&#380; promie&#324; okr&#281;gu wynosi jeden, z twierdzenia Pitagorasa otrzymujemy:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_1_2__editor_40078.svg' alt='40078_Equation_1_2_.svg' /></figure>Ta to&#380;samo&#347;&#263; stanowi podstaw&#281; do wyprowadzania dalszych to&#380;samo&#347;ci. Dziel&#261;c to&#380;samo&#347;&#263; pitagorejsk&#261; przez cos^2 &#952; (zak&#322;adaj&#261;c, &#380;e cos &#952; &#8800; 0), otrzymujemy:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_2_2__editor_40078.svg' alt='40078_Equation_2_2_.svg' /></figure>To&#380;samo&#347;&#263; pitagorejska odgrywa r&#243;wnie&#380; istotn&#261; rol&#281; w bardziej z&#322;o&#380;onych to&#380;samo&#347;ciach, takich jak wz&#243;r na dodawanie cosinus&#243;w. Jedn&#261; z metod jego wyprowadzenia s&#261; rozumowania geometryczne z u&#380;yciem punkt&#243;w na okr&#281;gu jednostkowym oraz wzoru na odleg&#322;o&#347;&#263;. W tym podej&#347;ciu wsp&#243;&#322;rz&#281;dne kluczowych punkt&#243;w wyra&#380;aj&#261; si&#281; kombinacjami warto&#347;ci sinusa i cosinusa. Po zastosowaniu wzoru na odleg&#322;o&#347;&#263; i odpowiednich uproszczeniach to&#380;samo&#347;&#263; pitagorejska redukuje wyra&#380;enia zawieraj&#261;ce kwadraty sk&#322;adnik&#243;w trygonometrycznych. Ostatecznie prowadzi to do:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_3_2__editor_40078.svg' alt='40078_Equation_3_2_.svg' /></figure>Zast&#281;puj&#261;c t przez &#8722;t we wzorze na dodawanie oraz korzystaj&#261;c z parzysto&#347;ci cosinusa i nieparzysto&#347;ci sinusa, otrzymujemy wz&#243;r na odejmowanie:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_4_2__editor_40078.svg' alt='40078_Equation_4_2_.svg' /></figure>Wzory na dodawanie i odejmowanie sinus&#243;w mo&#380;na r&#243;wnie&#380; wyprowadzi&#263;, stosuj&#261;c to&#380;samo&#347;ci wsp&#243;&#322;funkcyjne oraz przekszta&#322;cenia k&#261;t&#243;w:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_5_2__editor_40078.svg' alt='40078_Equation_5_2_.svg' /></figure>Wreszcie to&#380;samo&#347;ci na dodawanie i odejmowanie tangens&#243;w uzyskuje si&#281; ze wzor&#243;w na sum&#281; sinusa i cosinusa, bior&#261;c ich iloraz:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/450;' src='https://cloudfront.jove.com/files/ftp_upload/40078/40078_Equation_6_editor_40078.svg' alt='40078_Equation_6.svg' /></figure>To&#380;samo&#347;ci te s&#261; niezb&#281;dnymi narz&#281;dziami w upraszczaniu wyra&#380;e&#324; algebraicznych, rozwi&#261;zywaniu r&#243;wna&#324; trygonometrycznych oraz przy zastosowaniach rachunku r&#243;&#380;niczkowego i ca&#322;kowego.

Tożsamości trygonometryczne I

To&#380;samo&#347;ci trygonometryczne to r&#243;wnania wi&#261;&#380;&#261;ce funkcje trygonometryczne, kt&#243;re s&#261; spe&#322;nione dla wszystkich ...

Education

JoVE Core

Math Fundamentals

Podstawy matematyki

Unit

Jednostka

Trigonometry

Trygonometria

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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 Pythagorean identity. This result helps build later trig formulas and shows how one identity can generate others.

The Pythagorean identity also supports the cosine addition formula. One way to derive it uses points on the unit circle and the distance formula. The coordinates of the points involve sine and cosine terms. After simplifying the distance expression, the Pythagorean identity removes square terms and leads to the cosine sum formula.

The subtraction formula for cosine follows from the addition formula. Replacing t with -t and using the even and odd properties of cosine and sine gives the cosine difference identity. The sine addition and subtraction formulas can also be derived through cofunction identities and angle transformations.

The tangent addition and subtraction identities come from the sine and cosine sum formulas. Taking the ratio of the sine and cosine results produces the tangent formulas. These identities are useful for simplifying algebra, solving trigonometric equations, and working in calculus.

Unit Circle Trig Identities and Sum Formulas

unit-circle-trig-identities-and-sum-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

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 unit circle leads to Pythagorean, sine, cosine, and tangent identities, including addition and subtraction formulas for high school math.

Unit Circle Trig Identities and Sum 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

Unit Circle Trig Identities and Sum Formulas

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