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Fortgeschrittene trigonometrische funktionen

Sinussatz

Sinusfunktion einfach erklärt: Wie du Sinus und Kosinus im Taschenrechner benutzt

Name: Knowunity
Brand: Knowunity

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Sinusfunktion einfach erklärt: Wie du Sinus und Kosinus im Taschenrechner benutzt

Laura Sophie

@laurasophiestudies

7.576 Follower

The Sinusfunktion Formel (sine function formula) is a fundamental concept in trigonometry, describing oscillating behavior in mathematics and physics. This comprehensive guide explores the properties, parameters, and applications of sine functions, including how to graph them and use calculators for calculations.

• The sine function is defined on the unit circle, with y = sin(x) representing the y-coordinate of a point on the circle for a given angle x.
• Key properties include a domain of all real numbers, a range of [-1, 1], and a period of 2π.
• The general form of a sine function is y = a * sin(b(x - c)) + d, where a, b, c, and d are parameters affecting amplitude, frequency, phase shift, and vertical shift respectively.
• Graphing sine functions involves understanding how these parameters influence the shape and position of the curve.
• Modern calculators can compute sine values for angles in both degrees and radians, but proper mode settings are crucial for accurate results.

19.12.2021

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More Examples of Amplitude Changes

This page continues the discussion on amplitude changes in sine functions, providing additional examples and graphs.

Examples include: • y = f(x) = 0.8 sin(x) • y = f₅(x) = -2.5 sin(x)

For each function, the graph is shown alongside the standard sine function for comparison. The page emphasizes: • How the amplitude affects the range of the function • The relationship between the parameter a and the resulting amplitude

Highlight: Even when a is negative, the amplitude is still its absolute value, but the graph is inverted.

These examples help reinforce the concept of how the amplitude parameter in the Sinusfunktion Formel (sine function formula) directly impacts the function's graph and range.

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Amplitude Changes in Sine Functions

This page explores how changing the amplitude affects the graph of a sine function. The Allgemeine Sinusfunktion (general sine function) is introduced as y = f(x) = a · sin(x).

Key points: • The amplitude A is equal to the absolute value of a: A = |a| • When a > 0, the graph is unchanged in shape but stretched vertically • When a < 0, the graph is both stretched vertically and reflected over the x-axis

Example: For y = f₂(x) = 2 · sin(x), the amplitude is 2, and the range is [-2, 2]

Highlight: The amplitude is always positive, regardless of the sign of a.

The page includes graphs comparing the standard sine function (a = 1) with variations where a = 2, a = -0.5, and other values, illustrating how the amplitude parameter affects the function's shape and range.

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Frequency Changes in Sine Functions

This section introduces how changes to the frequency parameter b in the Allgemeine Sinusfunktion (general sine function) y = f(x) = sin(bx) affect the graph.

Key points: • The parameter b causes stretching or compression of the graph in the x-direction • It changes the period of the function: P = 2π/|b| • The amplitude and range remain unchanged

The page presents three examples:

y = f(x) = sin(2x)
y = f(x) = sin(½x)
y = f(x) = sin(⅔x)

For each case, the graph is shown and the new period is calculated.

Example: For sin(2x), the period is P = 2π/2 = π

Highlight: The frequency parameter b affects both the period and the locations of zeros (roots) of the function.

The relationship between b, the period, and the zeros is explained in detail, providing a comprehensive understanding of how this parameter influences the sine function's behavior.

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Properties of the Sine Function

This section delves into the key properties of the Sinusfunktion (sine function), which are essential for understanding its behavior and applications.

The main properties discussed are: • Domain: All real numbers (R) • Range: [-1, 1] • Amplitude: 1 (for the standard sine function) • Zeros (roots): x₀ = k · π, where k is any integer

Highlight: The sine function is periodic with a period of 2π.

The page also covers important trigonometric identities: • sin(α + k · 360°) = sin α • sin(-α) = -sin α • sin(180° - α) = sin α

Example: sin 30° = 0.5, and sin 150° = 0.5 (demonstrating the 180° - α identity)

These properties and identities are crucial for solving trigonometric equations and understanding the behavior of sine waves in various applications.

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Complex Sine Functions

This final page introduces more complex sine functions that combine multiple parameter changes. The general form y = a · sin(bx) is explored, where both amplitude and frequency are modified.

An example is provided: y = f(x) = -1.7 sin(⅙x)

The analysis includes: • Determining the amplitude: A = |-1.7| = 1.7 • Calculating the period: P = 2π/(⅙) = 12π • Noting that the negative coefficient results in a graph reflection

Highlight: When multiple parameters are changed, it's important to consider how each affects the graph independently and then combine these effects.

The page also includes a more complex example: y = f(x) = 3.2 sin(2x)

This example demonstrates how to handle cases where both the amplitude and frequency are altered significantly from the standard sine function.

Example: For 3.2 sin(2x), the amplitude is 3.2, and the period is π.

These complex examples help students understand how the Sinusfunktion Parameter interact to create a wide variety of wave forms, preparing them for more advanced applications in mathematics and physics.

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Introduction to Sine Functions

The sine function is introduced as a fundamental trigonometric function derived from the unit circle. Its graph is presented, showing the characteristic wave-like pattern.

Definition: The sine of angle α is defined as the y-coordinate of the point P(x,y) on the unit circle corresponding to the angle α.

Highlight: The sine function is unique (one-to-one) and is represented by the equation y = f(x) = sin x.

The page also covers: • How to calculate sine values using a calculator • The importance of setting the calculator to the correct angle mode (degrees or radians) • Examples of sine calculations for various angles

Example: sin 45° ≈ 0.710, sin 235° ≈ -0.849

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Angles and Equivalent Angles

This page focuses on understanding angles in both degree and radian measures, as well as the concept of equivalent angles.

Key topics covered include: • Drawing angles with specific measures, considering the direction of rotation • Finding equivalent angles within given intervals • Calculating sine values for various angles using a calculator

Example: For α = 42°, equivalent angles in the interval -850° ≤ α ≤ 700° are -348°, 42°, and 402°.

The page also provides exercises for: • Graphing the Sinusfunktion (sine function) over specified intervals • Completing a table of sine values for different angle measures

Highlight: Understanding equivalent angles is crucial for working with periodic functions like sine.

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Practice with Frequency Changes

This page provides practice problems focusing on sine functions with different frequency parameters.

Examples include:

y = f(x) = sin(π/2 · x)
y = f₁₂(x) = sin(2π/5 · x)
y = f(x) = sin(4/3 · x)

For each function, students are asked to: • Determine the period • Calculate the zeros (roots) • Sketch the graph

Highlight: The period is calculated using the formula P = 2π/|b|, where b is the frequency parameter.

The page also includes a step-by-step solution for finding zeros and extrema points, demonstrating how to use the period to determine these critical points on the graph.

Example: For sin(π/2 · x), the period is P = 2π/(π/2) = 4, and the zeros occur at x = 0, 2, 4, ...

This practice reinforces the relationship between the frequency parameter and the function's behavior, a crucial aspect of understanding the Sinusfunktion Parameter (sine function parameters).

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Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

Knowunity wurde bei Apple als "Featured Story" ausgezeichnet und hat die App-Store-Charts in der Kategorie Bildung in Deutschland, Italien, Polen, der Schweiz und dem Vereinigten Königreich regelmäßig angeführt. Werde noch heute Mitglied bei Knowunity und hilf Millionen von Schüler:innen auf der ganzen Welt.

Laden im

Google Play

Laden im

App Store

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

Durchschnittliche App-Bewertung

13 M

Schüler:innen lieben Knowunity

#1

In Bildungs-App-Charts in 12 Ländern

950 K+

Schüler:innen haben Lernzettel hochgeladen

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

iOS User

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Philipp, iOS User

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Lena, iOS Userin

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.

Fächer

Mathe

Funktionen und analysis

Fortgeschrittene trigonometrische funktionen

Sinussatz

Sinusfunktion einfach erklärt: Wie du Sinus und Kosinus im Taschenrechner benutzt

Laura Sophie

@laurasophiestudies

7.576 Follower

19.12.2021

12799

11/12

Mathe

338

More Examples of Amplitude Changes

This page continues the discussion on amplitude changes in sine functions, providing additional examples and graphs.

Examples include: • y = f(x) = 0.8 sin(x) • y = f₅(x) = -2.5 sin(x)

Highlight: Even when a is negative, the amplitude is still its absolute value, but the graph is inverted.

These examples help reinforce the concept of how the amplitude parameter in the Sinusfunktion Formel (sine function formula) directly impacts the function's graph and range.

Amplitude Changes in Sine Functions

This page explores how changing the amplitude affects the graph of a sine function. The Allgemeine Sinusfunktion (general sine function) is introduced as y = f(x) = a · sin(x).

Example: For y = f₂(x) = 2 · sin(x), the amplitude is 2, and the range is [-2, 2]

Highlight: The amplitude is always positive, regardless of the sign of a.

Frequency Changes in Sine Functions

This section introduces how changes to the frequency parameter b in the Allgemeine Sinusfunktion (general sine function) y = f(x) = sin(bx) affect the graph.

Key points: • The parameter b causes stretching or compression of the graph in the x-direction • It changes the period of the function: P = 2π/|b| • The amplitude and range remain unchanged

The page presents three examples:

y = f(x) = sin(2x)
y = f(x) = sin(½x)
y = f(x) = sin(⅔x)

For each case, the graph is shown and the new period is calculated.

Example: For sin(2x), the period is P = 2π/2 = π

Highlight: The frequency parameter b affects both the period and the locations of zeros (roots) of the function.

The relationship between b, the period, and the zeros is explained in detail, providing a comprehensive understanding of how this parameter influences the sine function's behavior.

Properties of the Sine Function

This section delves into the key properties of the Sinusfunktion (sine function), which are essential for understanding its behavior and applications.

The main properties discussed are: • Domain: All real numbers (R) • Range: [-1, 1] • Amplitude: 1 (for the standard sine function) • Zeros (roots): x₀ = k · π, where k is any integer

Highlight: The sine function is periodic with a period of 2π.

The page also covers important trigonometric identities: • sin(α + k · 360°) = sin α • sin(-α) = -sin α • sin(180° - α) = sin α

Example: sin 30° = 0.5, and sin 150° = 0.5 (demonstrating the 180° - α identity)

These properties and identities are crucial for solving trigonometric equations and understanding the behavior of sine waves in various applications.

Complex Sine Functions

This final page introduces more complex sine functions that combine multiple parameter changes. The general form y = a · sin(bx) is explored, where both amplitude and frequency are modified.

An example is provided: y = f(x) = -1.7 sin(⅙x)

The analysis includes: • Determining the amplitude: A = |-1.7| = 1.7 • Calculating the period: P = 2π/(⅙) = 12π • Noting that the negative coefficient results in a graph reflection

Highlight: When multiple parameters are changed, it's important to consider how each affects the graph independently and then combine these effects.

The page also includes a more complex example: y = f(x) = 3.2 sin(2x)

This example demonstrates how to handle cases where both the amplitude and frequency are altered significantly from the standard sine function.

Example: For 3.2 sin(2x), the amplitude is 3.2, and the period is π.

Introduction to Sine Functions

The sine function is introduced as a fundamental trigonometric function derived from the unit circle. Its graph is presented, showing the characteristic wave-like pattern.

Definition: The sine of angle α is defined as the y-coordinate of the point P(x,y) on the unit circle corresponding to the angle α.

Highlight: The sine function is unique (one-to-one) and is represented by the equation y = f(x) = sin x.

Example: sin 45° ≈ 0.710, sin 235° ≈ -0.849

Angles and Equivalent Angles

This page focuses on understanding angles in both degree and radian measures, as well as the concept of equivalent angles.

Example: For α = 42°, equivalent angles in the interval -850° ≤ α ≤ 700° are -348°, 42°, and 402°.

The page also provides exercises for: • Graphing the Sinusfunktion (sine function) over specified intervals • Completing a table of sine values for different angle measures

Highlight: Understanding equivalent angles is crucial for working with periodic functions like sine.

Practice with Frequency Changes

This page provides practice problems focusing on sine functions with different frequency parameters.

Examples include:

y = f(x) = sin(π/2 · x)
y = f₁₂(x) = sin(2π/5 · x)
y = f(x) = sin(4/3 · x)

For each function, students are asked to: • Determine the period • Calculate the zeros (roots) • Sketch the graph

Highlight: The period is calculated using the formula P = 2π/|b|, where b is the frequency parameter.

The page also includes a step-by-step solution for finding zeros and extrema points, demonstrating how to use the period to determine these critical points on the graph.

Example: For sin(π/2 · x), the period is P = 2π/(π/2) = 4, and the zeros occur at x = 0, 2, 4, ...

This practice reinforces the relationship between the frequency parameter and the function's behavior, a crucial aspect of understanding the Sinusfunktion Parameter (sine function parameters).

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

Knowunity wurde bei Apple als "Featured Story" ausgezeichnet und hat die App-Store-Charts in der Kategorie Bildung in Deutschland, Italien, Polen, der Schweiz und dem Vereinigten Königreich regelmäßig angeführt. Werde noch heute Mitglied bei Knowunity und hilf Millionen von Schüler:innen auf der ganzen Welt.

Laden im

Google Play

Laden im

App Store

4.9+

Durchschnittliche App-Bewertung

13 M

Schüler:innen lieben Knowunity

#1

In Bildungs-App-Charts in 12 Ländern

950 K+

Schüler:innen haben Lernzettel hochgeladen

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

iOS User

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Philipp, iOS User

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Lena, iOS Userin

Sinusfunktion einfach erklärt: Wie du Sinus und Kosinus im Taschenrechner benutzt

More Examples of Amplitude Changes

Amplitude Changes in Sine Functions

Frequency Changes in Sine Functions

Properties of the Sine Function

Complex Sine Functions

Introduction to Sine Functions

Angles and Equivalent Angles

Practice with Frequency Changes

Ähnliche Inhalte

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

13 M

#1

950 K+

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.

Sinusfunktion einfach erklärt: Wie du Sinus und Kosinus im Taschenrechner benutzt

More Examples of Amplitude Changes

Amplitude Changes in Sine Functions

Frequency Changes in Sine Functions

Properties of the Sine Function

Complex Sine Functions

Introduction to Sine Functions

Angles and Equivalent Angles

Practice with Frequency Changes

Ähnliche Inhalte

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

13 M

#1

950 K+

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.