Materie

Matematica

Geometria

Trigonometria

Teoremi sui triangoli rettangoli, della corda, dei seni,formula goniometrica per l'area di un triangolo qualsiasi

Impara la Circonferenza Goniometrica: Seno, Coseno e Tangente con Raggio Unitario

Name: Knowunity
Brand: Knowunity

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Impara la Circonferenza Goniometrica: Seno, Coseno e Tangente con Raggio Unitario

kami

@kami_basi

64 Follower

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The unit circle is a fundamental concept in trigonometry, used to define and visualize trigonometric functions. This comprehensive guide explores its key aspects, including the definitions of sine, cosine, and tangent, their graphical representations, and associated angles.

Key points:

The unit circle has a radius of 1 and is centered at the origin (0,0)
Trigonometric functions are defined using coordinates on the unit circle
Angles are measured in both degrees and radians
Graphs of sine, cosine, and tangent functions are periodic
Associated angles and their relationships are crucial for solving trigonometric problems

3/2/2023

6452

Goniometria
Una circonferenza goniometrica a il
centro in (0,0) e raggio 1.
↑(0:1)
(-1;0
(0;-1)
15/5/2
45°>11
90°<-> 11
P
X = 12°• 11:
180°

Vedi

Unit Circle Basics

The unit circle is the foundation of trigonometry, serving as a visual representation of trigonometric functions. It is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.

Definition: The circonferenza goniometrica (unit circle) is a circle with a radius of 1 and center at (0,0) in the coordinate plane.

Key points about the unit circle:

The circumference of the unit circle is 2π.
Angles are measured counterclockwise from the positive x-axis.
Coordinates on the unit circle represent cosine (x) and sine (y) values.

Highlight: The unit circle allows us to visualize trigonometric functions and their relationships.

Example: For an angle of 30°, the coordinates on the unit circle are (√3/2, 1/2), representing cos 30° and sin 30° respectively.

The guide provides a detailed diagram of the unit circle, showing key angle measurements in both degrees and radians.

Vocabulary: Raggio unitario significato refers to the unit radius, which is essential for defining trigonometric functions on the unit circle.

Vedi

Trigonometric Values Table

This section provides a comprehensive table of trigonometric values for common angles.

The table includes:

Angles in degrees and radians
Sine, cosine, and tangent values

Highlight: The Circonferenza goniometrica tabella offers a quick reference for important trigonometric values.

Key angles covered: 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°

Example: For 45°, sin 45° = cos 45° = √2/2, and tan 45° = 1

The page also emphasizes the periodicity of these functions and their relationships.

Vedi

Trigonometric Functions on the Unit Circle

This page delves into the definitions of cosine and sine using the unit circle, as well as introducing the concept of radians.

Key points:

Cosine (x-coordinate) and sine (y-coordinate) are defined using points on the unit circle.
One radian is the angle subtended by an arc length equal to the radius.
The relationship between degrees and radians is given: π radians = 180°.

Definition: Circonferenza goniometrica seno e coseno refers to the representation of sine and cosine on the unit circle.

Formula: sin²θ + cos²θ = 1 (Pythagorean identity)

The page also provides key values for sine and cosine at notable angles (0°, 90°, 180°, 270°, 360°).

Example: cos 0° = 1, sin 0° = 0; cos 90° = 0, sin 90° = 1

Vedi

Complementary Angles and Cotangent

This section explores the relationship between complementary angles and introduces the cotangent function.

Key points:

Complementary angles sum to π/2 (90°).
The sine of an angle equals the cosine of its complement.

Definition: Cotangent is defined as the reciprocal of tangent: cot θ = 1 / tan θ = cos θ / sin θ

The page includes a visual representation of complementary angles on the unit circle and formulas relating sine and cosine of complementary angles.

Formula: sin(π/2 - θ) = cos θ, cos(π/2 - θ) = sin θ

A table of cotangent values for common angles is provided, along with its graph.

Highlight: The Circonferenza goniometrica valori for cotangent complement those of sine, cosine, and tangent.

Vedi

Graphical Representations

This page presents the graphs of cosine, sine, and tangent functions.

Key features:

Cosine and sine graphs have a period of 2π.
Tangent graph has a period of π.
Cosine graph is shifted π/2 to the left compared to the sine graph.

Highlight: Grafico coseno, seno, coseno, tangente formule are visually represented, showing their periodic nature and key characteristics.

The graphs include important points such as x-intercepts, y-intercepts, and asymptotes (for tangent).

Example: The sine function has y-intercepts at 0°, 180°, and 360°, while reaching maximum values of 1 at 90° and minimum values of -1 at 270°.

Vedi

Quadrants and Tangent Function

This section explores the signs of trigonometric functions in different quadrants and introduces the tangent function.

Key points:

The signs of sine and cosine vary depending on the quadrant.
Tangent is defined as the ratio of sine to cosine: tan θ = sin θ / cos θ.

Definition: Circonferenza goniometrica tangente refers to the tangent function in relation to the unit circle.

The page includes diagrams showing the signs of sine and cosine in each quadrant, as well as the graphical representation of the tangent function on the unit circle.

Formula: cos θ = ±1 / √(1 + tan²θ), sin θ = ±tan θ / √(1 + tan²θ)

Vedi

More Associated Angles

This page continues the discussion of associated angles, focusing on angles that differ by π.

Key relationship covered:

Angles differing by π: (π + θ)

Formula: cos(π + θ) = -cos θ, sin(π + θ) = -sin θ, tan(π + θ) = tan θ

The guide provides examples and visual representations on the unit circle.

Example: sin 210° = sin(180° + 30°) = -sin 30° = -1/2

The page also revisits opposite angles (-θ) with additional examples.

Highlight: Understanding Angoli associati formule is crucial for solving complex trigonometric problems.

Vedi

Associated Angles

This section introduces the concept of associated angles and their relationships in trigonometry.

Key relationships covered:

Supplementary angles: (π - θ)
Opposite angles: (-θ)
Complementary angles: (π/2 - θ)

Definition: Angoli associati goniometria refers to angles that have specific relationships with a given angle θ.

For each relationship, the guide provides formulas for sine, cosine, and tangent.

Formula: For supplementary angles, cos(π - θ) = -cos θ, sin(π - θ) = sin θ, tan(π - θ) = -tan θ

Example: cos 120° = -cos(180° - 60°) = -cos 60° = -1/2

The page includes visual representations of these relationships on the unit circle.

Vedi

Inverse Trigonometric Functions

This page introduces inverse trigonometric functions, focusing on arcsine and arccosine.

Key points:

Inverse functions "undo" the original trigonometric functions.
The domain and range of inverse functions are restricted to ensure they are functions.

For arcsine:

Domain: [-1, 1]
Range: [-π/2, π/2]

For arccosine:

Domain: [-1, 1]
Range: [0, π]

Highlight: Graphs of inverse trigonometric functions are obtained by reflecting the original function over the line y = x and restricting the domain.

The page includes graphical representations of both arcsine and arccosine functions.

Example: arcsin(1/2) = π/6 (30°), as sin(π/6) = 1/2

Vedi

Inverse Trigonometric Functions (Continued)

This page continues the discussion of inverse trigonometric functions, focusing on arccosine.

Key points:

The arccosine function is the inverse of the cosine function with a restricted domain.
The graph of arccosine is obtained by reflecting the cosine graph over y = x and restricting the domain.

Definition: Arccosine (arccos x) gives the angle whose cosine is x, restricted to the range [0, π].

The page includes a detailed graph of the arccosine function, showing its domain and range.

Example: arccos(1/2) = π/3 (60°), as cos(π/3) = 1/2

Highlight: Understanding inverse trigonometric functions is crucial for solving equations involving trigonometric functions.

The guide emphasizes the importance of domain and range restrictions in defining these inverse functions.

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

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Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

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Studenti che usano Knowunity

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Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS

Adoro questa app ❤️, la uso praticamente sempre quando studio.

Materie

Matematica

Geometria

Trigonometria

Teoremi sui triangoli rettangoli, della corda, dei seni,formula goniometrica per l'area di un triangolo qualsiasi

Impara la Circonferenza Goniometrica: Seno, Coseno e Tangente con Raggio Unitario

kami

@kami_basi

64 Follower

Segui

Key points:

The unit circle has a radius of 1 and is centered at the origin (0,0)
Trigonometric functions are defined using coordinates on the unit circle
Angles are measured in both degrees and radians
Graphs of sine, cosine, and tangent functions are periodic
Associated angles and their relationships are crucial for solving trigonometric problems

3/2/2023

6452

4ªl

Matematica

333

Unit Circle Basics

Definition: The circonferenza goniometrica (unit circle) is a circle with a radius of 1 and center at (0,0) in the coordinate plane.

Key points about the unit circle:

The circumference of the unit circle is 2π.
Angles are measured counterclockwise from the positive x-axis.
Coordinates on the unit circle represent cosine (x) and sine (y) values.

Highlight: The unit circle allows us to visualize trigonometric functions and their relationships.

Example: For an angle of 30°, the coordinates on the unit circle are (√3/2, 1/2), representing cos 30° and sin 30° respectively.

The guide provides a detailed diagram of the unit circle, showing key angle measurements in both degrees and radians.

Vocabulary: Raggio unitario significato refers to the unit radius, which is essential for defining trigonometric functions on the unit circle.

Trigonometric Values Table

This section provides a comprehensive table of trigonometric values for common angles.

The table includes:

Angles in degrees and radians
Sine, cosine, and tangent values

Highlight: The Circonferenza goniometrica tabella offers a quick reference for important trigonometric values.

Key angles covered: 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°

Example: For 45°, sin 45° = cos 45° = √2/2, and tan 45° = 1

The page also emphasizes the periodicity of these functions and their relationships.

Trigonometric Functions on the Unit Circle

This page delves into the definitions of cosine and sine using the unit circle, as well as introducing the concept of radians.

Key points:

Cosine (x-coordinate) and sine (y-coordinate) are defined using points on the unit circle.
One radian is the angle subtended by an arc length equal to the radius.
The relationship between degrees and radians is given: π radians = 180°.

Definition: Circonferenza goniometrica seno e coseno refers to the representation of sine and cosine on the unit circle.

Formula: sin²θ + cos²θ = 1 (Pythagorean identity)

The page also provides key values for sine and cosine at notable angles (0°, 90°, 180°, 270°, 360°).

Example: cos 0° = 1, sin 0° = 0; cos 90° = 0, sin 90° = 1

Complementary Angles and Cotangent

This section explores the relationship between complementary angles and introduces the cotangent function.

Key points:

Complementary angles sum to π/2 (90°).
The sine of an angle equals the cosine of its complement.

Definition: Cotangent is defined as the reciprocal of tangent: cot θ = 1 / tan θ = cos θ / sin θ

The page includes a visual representation of complementary angles on the unit circle and formulas relating sine and cosine of complementary angles.

Formula: sin(π/2 - θ) = cos θ, cos(π/2 - θ) = sin θ

A table of cotangent values for common angles is provided, along with its graph.

Highlight: The Circonferenza goniometrica valori for cotangent complement those of sine, cosine, and tangent.

Graphical Representations

This page presents the graphs of cosine, sine, and tangent functions.

Key features:

Cosine and sine graphs have a period of 2π.
Tangent graph has a period of π.
Cosine graph is shifted π/2 to the left compared to the sine graph.

Highlight: Grafico coseno, seno, coseno, tangente formule are visually represented, showing their periodic nature and key characteristics.

The graphs include important points such as x-intercepts, y-intercepts, and asymptotes (for tangent).

Example: The sine function has y-intercepts at 0°, 180°, and 360°, while reaching maximum values of 1 at 90° and minimum values of -1 at 270°.

Quadrants and Tangent Function

This section explores the signs of trigonometric functions in different quadrants and introduces the tangent function.

Key points:

The signs of sine and cosine vary depending on the quadrant.
Tangent is defined as the ratio of sine to cosine: tan θ = sin θ / cos θ.

Definition: Circonferenza goniometrica tangente refers to the tangent function in relation to the unit circle.

The page includes diagrams showing the signs of sine and cosine in each quadrant, as well as the graphical representation of the tangent function on the unit circle.

Formula: cos θ = ±1 / √(1 + tan²θ), sin θ = ±tan θ / √(1 + tan²θ)

More Associated Angles

This page continues the discussion of associated angles, focusing on angles that differ by π.

Key relationship covered:

Angles differing by π: (π + θ)

Formula: cos(π + θ) = -cos θ, sin(π + θ) = -sin θ, tan(π + θ) = tan θ

The guide provides examples and visual representations on the unit circle.

Example: sin 210° = sin(180° + 30°) = -sin 30° = -1/2

The page also revisits opposite angles (-θ) with additional examples.

Highlight: Understanding Angoli associati formule is crucial for solving complex trigonometric problems.

Associated Angles

This section introduces the concept of associated angles and their relationships in trigonometry.

Key relationships covered:

Supplementary angles: (π - θ)
Opposite angles: (-θ)
Complementary angles: (π/2 - θ)

Definition: Angoli associati goniometria refers to angles that have specific relationships with a given angle θ.

For each relationship, the guide provides formulas for sine, cosine, and tangent.

Formula: For supplementary angles, cos(π - θ) = -cos θ, sin(π - θ) = sin θ, tan(π - θ) = -tan θ

Example: cos 120° = -cos(180° - 60°) = -cos 60° = -1/2

The page includes visual representations of these relationships on the unit circle.

Inverse Trigonometric Functions

This page introduces inverse trigonometric functions, focusing on arcsine and arccosine.

Key points:

Inverse functions "undo" the original trigonometric functions.
The domain and range of inverse functions are restricted to ensure they are functions.

For arcsine:

Domain: [-1, 1]
Range: [-π/2, π/2]

For arccosine:

Domain: [-1, 1]
Range: [0, π]

Highlight: Graphs of inverse trigonometric functions are obtained by reflecting the original function over the line y = x and restricting the domain.

The page includes graphical representations of both arcsine and arccosine functions.

Example: arcsin(1/2) = π/6 (30°), as sin(π/6) = 1/2

Inverse Trigonometric Functions (Continued)

This page continues the discussion of inverse trigonometric functions, focusing on arccosine.

Key points:

The arccosine function is the inverse of the cosine function with a restricted domain.
The graph of arccosine is obtained by reflecting the cosine graph over y = x and restricting the domain.

Definition: Arccosine (arccos x) gives the angle whose cosine is x, restricted to the range [0, π].

The page includes a detailed graph of the arccosine function, showing its domain and range.

Example: arccos(1/2) = π/3 (60°), as cos(π/3) = 1/2

Highlight: Understanding inverse trigonometric functions is crucial for solving equations involving trigonometric functions.

The guide emphasizes the importance of domain and range restrictions in defining these inverse functions.

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Non c'è niente di adatto? Esplorare altre aree tematiche.

Knowunity è l'app per l'istruzione numero 1 in cinque paesi europei

Knowunity è stata inserita in un articolo di Apple ed è costantemente in cima alle classifiche degli app store nella categoria istruzione in Germania, Italia, Polonia, Svizzera e Regno Unito. Unisciti a Knowunity oggi stesso e aiuta milioni di studenti in tutto il mondo.

Scarica

Google Play

Scarica

App Store

4.9+

Valutazione media dell'app

13 M

Studenti che usano Knowunity

#1

Nelle classifiche delle app per l'istruzione in 12 Paesi

950 K+

Studenti che hanno caricato appunti

Non siete ancora sicuri? Guarda cosa dicono gli altri studenti...

Utente iOS

Adoro questa applicazione [...] consiglio Knowunity a tutti!!! Sono passato da un 5 a una 8 con questa app

Stefano S, utente iOS

L'applicazione è molto semplice e ben progettata. Finora ho sempre trovato quello che stavo cercando

Susanna, utente iOS