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

Der Einheitskreis, der am Ursprung zentriert ist, definiert die trigonometrischen Funktionen.Jede reelle Zahl stellt eine Bogenlänge dar, die von der positiven x-Achse aus gemessen wird, und dreht sie gegen den Uhrzeigersinn um den Kreis.Die x-Koordinate ergibt den Kosinus der reellen Zahl. In ähnlicher Weise ergibt die y-Koordinate den Sinus.Der Tangente wird abgeleitet als Sinus über Kosinus, der Kotangens ist der Kosinus über den Sinus, der Sekans ist der Kehrwert des Kosinus und der Kosekans ist der Kehrwert des Sinus.Das Unterteilen des Kreises in vier Quadranten hilft dabei, das Vorzeichen einer Funktion zu identifizieren.Alle Funktionen sind im ersten Quadranten positiv, wobei x und y positiv sind. Da x negativ ist, ist der Kosinus im zweiten negativ. Da y positiv ist, ist auch Sinus positiv. Die Tangente, die y über x ist, ist negativ, weil ein Positiv über einem Negativen negativ negativ ist.Im dritten Fall sind Tangens und Kotangens positiv, während x und y negativ sind. Im vierten sind nur Kosinus und Sekante positiv, da x positiv und y negativ ist.Trigonometrische Funktionen sind nützlich für reale Berechnungen, z. B. die Schätzung der Höhe eines Gebäudes durch Multiplikation des bekannten horizontalen Abstands mit der Tangente des bekannten Neigungswinkels.

trigonometrische funktionen reeller zahlen

Trigonometric Functions of Real Numbers

Der Einheitskreis &#8211; ein Kreis mit Radius eins und Mittelpunkt im Ursprung des Koordinatensystems &#8211; dient als Grundlage f&#252;r die Definition trigonometrischer Funktionen. Die Bogenl&#228;nge bezeichnet in diesem Zusammenhang die entlang des Kreisumfangs gemessene Entfernung zwischen zwei Punkten und erm&#246;glicht die geometrische Darstellung reeller Zahlen. Jede reelle Zahl t entspricht einer Bogenl&#228;nge, die von der positiven x-Achse aus gegen den Uhrzeigersinn entlang des Kreisumfangs gemessen wird. Die Koordinaten eines Punktes auf dem Einheitskreis (x, y) bei einer gegebenen Bogenl&#228;nge definieren den Kosinus bzw. den Sinus der reellen Zahl, wobei gilt: x = cos t und y = sin t.Andere trigonometrische Funktionen leiten sich direkt von Sinus und Kosinus ab. Die Tangensfunktion ist definiert als das Verh&#228;ltnis von Sinus zu Kosinus:<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>Umgekehrt ist die Kotangensfunktion das Verh&#228;ltnis von Kosinus zu Sinus:<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>Die Sekans- und die Kosekansfunktion sind die Kehrwerte von Kosinus bzw. Sinus:<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>Die Aufteilung des Einheitskreises in vier Quadranten verdeutlicht die Vorzeichen der trigonometrischen Funktionen. Im ersten Quadranten sind alle Funktionen positiv. Im zweiten Quadranten nehmen nur Sinus und Kosekans positive Werte an. Im dritten Quadranten sind Tangens und Kotangens positiv, w&#228;hrend Kosinus und Sekans im vierten Quadranten positiv sind. Diese Vorzeichenregeln sind f&#252;r die Auswertung trigonometrischer Funktionen &#252;ber verschiedene Winkel hinweg unerl&#228;sslich.Die Symmetrie des Einheitskreises verleiht den Funktionen gerade bzw. ungerade Symmetrieeigenschaften. Kosinus und Sekans sind gerade Funktionen und erf&#252;llen cos (&#8722;t) = cos t und sec (&#8722;t) = sec t. Im Gegensatz dazu sind Sinus, Tangens, Kotangens und Kosekans ungerade Funktionen und erf&#252;llen Identit&#228;ten wie sin (&#8722;t) = &#8722;sin t und tan (&#8722;t) = &#8722;tan t.Zusammen bieten diese Definitionen und Vorzeichenkonventionen einen schl&#252;ssigen Rahmen f&#252;r das Verst&#228;ndnis des Verhaltens trigonometrischer Funktionen. Der Einheitskreis vermittelt nicht nur geometrische Intuition, sondern erleichtert zudem pr&#228;zise Berechnungen und Analysen und ist daher sowohl in der reinen als auch in der angewandten Mathematik unverzichtbar.

Trigonometrische Funktionen reeller Zahlen

Der Einheitskreis &#8211; ein Kreis mit Radius eins und Mittelpunkt im Ursprung des Koordinatensystems &#8211; dient als Grundlage f&#252;r die Definition ...

Education

Lehre

JoVE Core

Math Fundamentals

Mathematische Grundlagen

Unit

Einheit

Trigonometry

Trigonometrie

KEY TERMS AND CONCEPTS

WICHTIGE BEGRIFFE UND KONZEPTE

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...

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 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

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

Description

Transcript

Tags