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Fortgeschrittene trigonometrische funktionen

Sinus, Cosinus und Tangens einfach erklärt: Formeln, Rechenregeln und Eselsbrücken

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Sinus, Cosinus und Tangens einfach erklärt: Formeln, Rechenregeln und Eselsbrücken

Stella

@stella_lernzettel

1.654 Follower

Trigonometry explores the relationships between angles and sides in triangles, with a focus on sinus, cosinus, and tangens functions. These fundamental concepts are crucial for understanding circular motion, wave patterns, and various mathematical applications.

The unit circle provides a visual representation of trigonometric functions
Radian measure offers an alternative to degree measure for angles
Sine and cosine graphs exhibit periodic behavior with key characteristics
General sine and cosine functions can be modified using parameters

27.9.2021

7983

1.
2.
TRIGONOMETRIE
SINUS UND KOSINUS IM RECHTWINKLIGEN DREIECK
Sin
=
Gegenkathete
Hypotenuse
a
A
Sin +
COS -
Kathete
III. Quadrant
Sin -
CO

Visualization in the Unit Circle

The unit circle serves as a powerful tool for visualizing sinus and cosinus functions. This section explains how these trigonometric ratios can be represented on a circular graph.

Highlight: The y-coordinate of a point on the unit circle represents the sine value, while the x-coordinate represents the cosine value.

Key points covered in this section include:

The unit circle has a radius of 1, simplifying trigonometric calculations
Points on the circle are represented as P(x|y) = P(cos(α), sin(α))
The four quadrants of the circle correspond to different sign combinations for sine and cosine values

Example: In the first quadrant (0° to 90°), both sine and cosine are positive.

This visualization helps students understand the periodic nature of trigonometric functions and their behavior across different angle measures.

Öffnen

General Sine and Cosine Functions

The final section introduces the general forms of sine and cosine functions, which allow for modifications to the basic trigonometric graphs.

The general forms are:

f(x) = a · sin(b(x - c)) + d f(x) = a · cos(b(x - c)) + d

Each parameter has a specific effect on the graph:

a: Affects the amplitude and vertical stretch/compression
b: Affects the period and horizontal stretch/compression
c: Causes a horizontal shift
d: Causes a vertical shift

Vocabulary: Amplitude is the maximum displacement from the midline of a periodic function.

Example: In f(x) = 2sin(3x - π) + 1, the amplitude is 2, the period is 2π/3, there's a horizontal shift of π/3 to the right, and a vertical shift of 1 unit up.

Understanding these parameters allows for modeling a wide range of periodic phenomena and solving complex trigonometric problems.

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Sinus and Cosinus in Right-Angled Triangles

This section introduces the fundamental trigonometric ratios of sinus and cosinus in right-angled triangles. The relationships between the sides of a right-angled triangle are explained using these ratios.

Definition: Sinus (sin) is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Definition: Cosinus (cos) is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

These definitions provide the foundation for understanding trigonometric functions and their applications in various mathematical and real-world contexts.

Öffnen

Graphs of Sine and Cosine Functions

This section presents the graphical representations of the sine and cosine functions. The key characteristics of these graphs are discussed in detail.

Highlight: The sine and cosine functions are periodic with a period of 2π (or 360°).

Important features of the graphs include:

The amplitude of both functions is 1
The sine function is odd (point-symmetric), while the cosine function is even (axis-symmetric)
The cosine function is a horizontal shift of the sine function by π/2 (or 90°)

Example: sin(0°) = 0, cos(0°) = 1

Understanding these graphs is essential for analyzing periodic phenomena and solving trigonometric equations.

Öffnen

Degree and Radian Measure

This section introduces the concept of radian measure as an alternative to degree measure for angles. The relationship between these two systems is explained.

Definition: The radian measure of an angle is defined as the length of the arc on a unit circle subtended by that angle.

The conversion formula between degrees and radians is provided:

x = α · π / 180° α = x · 180° / π

Where x is the radian measure and α is the degree measure.

Example: 180° is equivalent to π radians.

Understanding both degree and radian measures is crucial for working with trigonometric functions and their applications in various fields of mathematics and physics.

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Fächer

Mathe

Funktionen und analysis

Fortgeschrittene trigonometrische funktionen

Sinus, Cosinus und Tangens einfach erklärt: Formeln, Rechenregeln und Eselsbrücken

Stella

@stella_lernzettel

1.654 Follower

The unit circle provides a visual representation of trigonometric functions
Radian measure offers an alternative to degree measure for angles
Sine and cosine graphs exhibit periodic behavior with key characteristics
General sine and cosine functions can be modified using parameters

27.9.2021

7983

Mathe

276

Visualization in the Unit Circle

The unit circle serves as a powerful tool for visualizing sinus and cosinus functions. This section explains how these trigonometric ratios can be represented on a circular graph.

Highlight: The y-coordinate of a point on the unit circle represents the sine value, while the x-coordinate represents the cosine value.

Key points covered in this section include:

The unit circle has a radius of 1, simplifying trigonometric calculations
Points on the circle are represented as P(x|y) = P(cos(α), sin(α))
The four quadrants of the circle correspond to different sign combinations for sine and cosine values

Example: In the first quadrant (0° to 90°), both sine and cosine are positive.

This visualization helps students understand the periodic nature of trigonometric functions and their behavior across different angle measures.

General Sine and Cosine Functions

The final section introduces the general forms of sine and cosine functions, which allow for modifications to the basic trigonometric graphs.

The general forms are:

f(x) = a · sin(b(x - c)) + d f(x) = a · cos(b(x - c)) + d

Each parameter has a specific effect on the graph:

a: Affects the amplitude and vertical stretch/compression
b: Affects the period and horizontal stretch/compression
c: Causes a horizontal shift
d: Causes a vertical shift

Vocabulary: Amplitude is the maximum displacement from the midline of a periodic function.

Example: In f(x) = 2sin(3x - π) + 1, the amplitude is 2, the period is 2π/3, there's a horizontal shift of π/3 to the right, and a vertical shift of 1 unit up.

Understanding these parameters allows for modeling a wide range of periodic phenomena and solving complex trigonometric problems.

Sinus and Cosinus in Right-Angled Triangles

Definition: Sinus (sin) is defined as the ratio of the opposite side to the hypotenuse in a right-angled triangle.

Definition: Cosinus (cos) is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

These definitions provide the foundation for understanding trigonometric functions and their applications in various mathematical and real-world contexts.

Graphs of Sine and Cosine Functions

This section presents the graphical representations of the sine and cosine functions. The key characteristics of these graphs are discussed in detail.

Highlight: The sine and cosine functions are periodic with a period of 2π (or 360°).

Important features of the graphs include:

The amplitude of both functions is 1
The sine function is odd (point-symmetric), while the cosine function is even (axis-symmetric)
The cosine function is a horizontal shift of the sine function by π/2 (or 90°)

Example: sin(0°) = 0, cos(0°) = 1

Understanding these graphs is essential for analyzing periodic phenomena and solving trigonometric equations.

Degree and Radian Measure

This section introduces the concept of radian measure as an alternative to degree measure for angles. The relationship between these two systems is explained.

Definition: The radian measure of an angle is defined as the length of the arc on a unit circle subtended by that angle.

The conversion formula between degrees and radians is provided:

x = α · π / 180° α = x · 180° / π

Where x is the radian measure and α is the degree measure.

Example: 180° is equivalent to π radians.

Understanding both degree and radian measures is crucial for working with trigonometric functions and their applications in various fields of mathematics and physics.