Name: unit circle trig: signs and symmetry
Uploaded: 2025-11-26T05:44:36.000Z
Duration: 1 min 30 s
Channel: JoVE

Okrąg jednostkowy, wyśrodkowany w punkcie początkowym, definiuje funkcje trygonometryczne.Każda liczba rzeczywista reprezentuje długość łuku mierzoną od dodatniej osi x i obracaj ją w kierunku przeciwnym do ruchu wskazówek zegara wokół okręgu.Współrzędna x daje cosinus liczby rzeczywistej. Podobnie, współrzędna y daje sinus.Styczna jest wyprowadzana jako sinus nad cosinusem, cotangens to cosinus nad sinusem, secans jest odwrotnością cosinusa, a cosecans jest odwrotnością sinusa.Podzielenie okręgu na cztery ćwiartki pomaga zidentyfikować znak funkcji.Wszystkie funkcje są dodatnie w pierwszej ćwiartce, gdzie x i y są dodatnie. W drugim, ponieważ x jest ujemne, cosinus jest ujemny. Ponieważ y jest dodatnie, sinus jest również dodatni. Styczna, która jest y nad x, jest ujemna, ponieważ dodatnia nad ujemną jest ujemna.W trzecim tangens i cotangens są dodatnie, ponieważ x i y są ujemne. W czwartym tylko cosinus i sieczność są dodatnie, ponieważ x jest dodatnie, a y jest ujemne.Funkcje trygonometryczne są przydatne w obliczeniach rzeczywistych, takich jak szacowanie wysokości budynku przez pomnożenie znanej odległości poziomej przez styczną znanego kąta nachylenia.

funkcje trygonometryczne liczb rzeczywistych

Trigonometric Functions of Real Numbers

Okr&#261;g jednostkowy &#8212; okr&#261;g o promieniu 1, kt&#243;rego &#347;rodek znajduje si&#281; w pocz&#261;tku uk&#322;adu wsp&#243;&#322;rz&#281;dnych &#8212; stanowi podstawowy uk&#322;ad odniesienia do definiowania funkcji trygonometrycznych. W tym kontek&#347;cie d&#322;ugo&#347;&#263; &#322;uku oznacza odleg&#322;o&#347;&#263; mierzona wzd&#322;u&#380; obwodu okr&#281;gu mi&#281;dzy dwoma punktami i stanowi spos&#243;b geometrycznego przedstawienia liczb rzeczywistych. Ka&#380;dej liczbie rzeczywistej t odpowiada d&#322;ugo&#347;&#263; &#322;uku odmierzana przeciwnie do ruchu wskaz&#243;wek zegara od dodatniej osi x wok&#243;&#322; okr&#281;gu. Wsp&#243;&#322;rz&#281;dne punktu (x, y) na okr&#281;gu jednostkowym dla danej d&#322;ugo&#347;ci &#322;uku definiuj&#261; odpowiednio cosinus i sinus tej liczby: x = cos t oraz y = sin t.Inne funkcje trygonometryczne wyprowadza si&#281; bezpo&#347;rednio z sinusa i cosinusa. Funkcja tangens jest zdefiniowana jako iloraz sinusa do cosinusa:<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/40073/40073_Equation_1_1__editor_40073.svg' alt='40073_Equation_1_1_.svg' /></figure>Natomiast funkcja cotangens to iloraz cosinusa do sinusa:<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/40073/40073_Equation_2_1__editor_40073.svg' alt='40073_Equation_2_1_.svg' /></figure>Sekans i kosekans s&#261; odpowiednio odwrotno&#347;ciami cosinusa i sinusa:<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/40073/40073_Equation_3_1__editor_40073.svg' alt='40073_Equation_3_1_.svg' /></figure>Podzia&#322; okr&#281;gu jednostkowego na cztery &#263;wiartki pozwala okre&#347;li&#263; znaki funkcji trygonometrycznych. W pierwszej &#263;wiartce wszystkie funkcje s&#261; dodatnie. W drugiej &#263;wiartce dodatnie pozostaj&#261; jedynie sinus i kosekans. W trzeciej &#263;wiartce dodatnie s&#261; tangens i cotangens, natomiast w czwartej &#8212; cosinus i sekans. Regu&#322;y znak&#243;w s&#261; niezb&#281;dne do poprawnego wyznaczania warto&#347;ci funkcji trygonometrycznych dla r&#243;&#380;nych k&#261;t&#243;w.Symetria okr&#281;gu jednostkowego nadaje funkcjom w&#322;asno&#347;ci parzysto&#347;ci lub nieparzysto&#347;ci. Cosinus i sekans s&#261; funkcjami parzystymi, to znaczy, &#380;e spe&#322;niaj&#261; zale&#380;no&#347;ci cos (&#8722;t) = cos t oraz sec (&#8722;t) = sec t. Z kolei sinus, tangens, cotangens i kosekans s&#261; funkcjami nieparzystymi, co wyra&#380;a si&#281; m.in. to&#380;samo&#347;ciami sin (&#8722;t) = &#8722;sin t oraz tan (&#8722;t) = &#8722;tan t.Powy&#380;sze definicje i konwencje znakowe tworz&#261; sp&#243;jne ramy dla zrozumienia zachowania funkcji trygonometrycznych. Okr&#261;g jednostkowy dostarcza nie tylko intuicji geometrycznej, lecz tak&#380;e u&#322;atwia precyzyjne obliczenia i analiz&#281;, co czyni go narz&#281;dziem niezb&#281;dnym zar&#243;wno w matematyce czystej, jak i stosowanej.

Funkcje trygonometryczne liczb rzeczywistych

Okr&#261;g jednostkowy &#8212; okr&#261;g o promieniu 1, kt&#243;rego &#347;rodek znajduje si&#281; w pocz&#261;tku uk&#322;adu ...

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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 equals sin t. The other trig functions come from these two values. Tangent is the ratio of sine to cosine. Cotangent is the ratio of cosine to sine. Secant is the reciprocal of cosine, and cosecant is the reciprocal of sine.

The four quadrants of the unit circle help show where each trigonometric function is positive or negative. In the first quadrant, all trigonometric functions are positive. In the second quadrant, only sine and cosecant are positive. In the third quadrant, tangent and cotangent are positive. In the fourth quadrant, cosine and secant are positive. These sign patterns are important when evaluating trig values for different angles.

The unit circle also shows symmetry in trigonometric functions. Cosine and secant are even functions, so cos(-t) = cos t and sec(-t) = sec t. Sine, tangent, cotangent, and cosecant are odd functions. For example, sin(-t) = -sin t and tan(-t) = -tan t. Together, these rules give a clear way to study trig functions and compute them across the coordinate plane.

Unit Circle Trig: Signs and Symmetry

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

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...

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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...

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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...

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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...

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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...

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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...

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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...

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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...

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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...

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Solving Trigonometric Equations

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key terms and concepts

Explore how the unit circle defines sine, cosine, tangent, and related functions, plus quadrant signs and even-odd symmetry for real numbers.

Unit Circle Trig: Signs and Symmetry | 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: Signs and Symmetry

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