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Mathe

Funktionen und analysis

Fortgeschrittene trigonometrische funktionen

Sinussatz

Easy Guide to Sinus, Cosinus & Tangens

Name: Knowunity
Brand: Knowunity

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Sophie♡︎

@sophie_btw

917 Follower

Trigonometry in right-angled triangles: Understanding sin, cos, and tan

This document covers key concepts in trigonometry, focusing on sin, cos, tan formulas and their applications in solving problems involving right-angled triangles. It includes:

Definitions and relationships between sine, cosine, and tangent
Calculating angles and sides using sin cos tan formulas
Solving real-world problems using trigonometric ratios
Advanced trigonometric concepts and proofs

Key points:

Sine, cosine, and tangent are ratios of sides in right-angled triangles
These ratios can be used to find unknown angles or side lengths
Trigonometric functions have applications in various fields, including physics and engineering
Understanding the unit circle helps visualize trigonometric relationships

23.3.2021

2351

Introduction to Trigonometric Ratios

This page introduces the fundamental trigonometric ratios: sine, cosine, and tangent. It presents a test paper with various problems involving these concepts.

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

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

Definition: Tangent (tan) is the ratio of the opposite side to the adjacent side in a right-angled triangle.

The page includes problems that require students to:

List relationships between sin, cos, and tan
Solve a real-world problem involving a fallen tree
Calculate angles in a triangle given its side lengths
Solve complex trigonometric problems using a given diagram
Analyze trigonometric functions for specific angle values

Example: A tree problem is presented where students must calculate the original height of a tree that has been bent by a storm, forming a 60° angle with the ground.

Klassenarbeit Nr. 2
Aufgabe 1:
Nenne vier verschiedene Beziehungen zwischen Sinus, Kosinus und/oder Tangens.
Sportklasse 10 b
Aufgabe 2: Ein

Review and Advanced Trigonometric Proofs

The final page provides corrections and additional explanations for the test problems, along with some advanced trigonometric proofs and identities.

Highlight: The page includes a proof of the identity (sin(x) + cos(x))² = 1 + 2sin(α)cos(α), demonstrating the interconnectedness of trigonometric functions.

Key points:

Corrections for previous calculations, emphasizing precision in trigonometric problem-solving
Advanced trigonometric identities and their proofs
Graphical representations of sine and cosine functions

Vocabulary: Trigonometric identity: An equation involving trigonometric functions that is true for all values of the variables involved.

The page concludes with graphical representations of sine and cosine functions, reinforcing the visual understanding of these trigonometric concepts and their periodic nature.

Öffnen

Trigonometric Functions and the Unit Circle

This page delves into more advanced concepts related to trigonometric functions and their representation on the unit circle.

Definition: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system, used to visualize trigonometric functions.

Key topics covered include:

Solving equations involving sine and cosine for specific angle ranges
Analyzing the behavior of trigonometric functions over extended angle ranges (0° to 720°)
Visualizing trigonometric functions on the unit circle and as graphs

Example: The page includes a problem asking for which angles between 0° and 720° the equation sin α = 0.75 is true, requiring students to consider multiple rotations around the unit circle.

The solutions demonstrate how to use the unit circle to find multiple angle solutions and how to interpret trigonometric functions graphically.

Öffnen

Advanced Trigonometric Concepts

This page continues with solutions to more complex trigonometric problems, including those involving multiple triangles and advanced trigonometric identities.

Vocabulary: Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

The solutions cover:

Calculating distances and heights using trigonometric ratios
Finding areas of triangles using trigonometric functions
Applying trigonometric identities to solve equations

Highlight: One problem involves calculating the area of a triangle using the formula A = ½ * base * height, where the height is found using the sine function.

The page also includes partial solutions and teacher annotations, indicating areas where students may have made errors or where additional explanation might be needed.

Öffnen

Solving Trigonometric Problems

This page demonstrates solutions to the problems presented in the test paper. It showcases various techniques for solving trigonometric equations and real-world applications.

Highlight: The solution to the tree problem shows how to use the sine function to calculate the tree's original height, which is approximately 8.66 meters.

The page includes detailed calculations for:

Using sin, cos, tan to calculate angles in a triangle
Applying the law of sines and cosines
Solving for unknown side lengths in complex triangles

Example: In one solution, the sine law is used to find an angle: sin α / a = sin β / b, where α and β are angles, and a and b are the opposite sides.

The solutions demonstrate the step-by-step process of solving trigonometric problems, emphasizing the importance of proper formula application and algebraic manipulation.

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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 11 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.

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 11 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

Easy Guide to Sinus, Cosinus & Tangens

Sophie♡︎

@sophie_btw

917 Follower

Trigonometry in right-angled triangles: Understanding sin, cos, and tan

This document covers key concepts in trigonometry, focusing on sin, cos, tan formulas and their applications in solving problems involving right-angled triangles. It includes:

Definitions and relationships between sine, cosine, and tangent
Calculating angles and sides using sin cos tan formulas
Solving real-world problems using trigonometric ratios
Advanced trigonometric concepts and proofs

Key points:

Sine, cosine, and tangent are ratios of sides in right-angled triangles
These ratios can be used to find unknown angles or side lengths
Trigonometric functions have applications in various fields, including physics and engineering
Understanding the unit circle helps visualize trigonometric relationships

23.3.2021

2351

Mathe

Introduction to Trigonometric Ratios

This page introduces the fundamental trigonometric ratios: sine, cosine, and tangent. It presents a test paper with various problems involving these concepts.

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

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

Definition: Tangent (tan) is the ratio of the opposite side to the adjacent side in a right-angled triangle.

The page includes problems that require students to:

List relationships between sin, cos, and tan
Solve a real-world problem involving a fallen tree
Calculate angles in a triangle given its side lengths
Solve complex trigonometric problems using a given diagram
Analyze trigonometric functions for specific angle values

Example: A tree problem is presented where students must calculate the original height of a tree that has been bent by a storm, forming a 60° angle with the ground.

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Review and Advanced Trigonometric Proofs

The final page provides corrections and additional explanations for the test problems, along with some advanced trigonometric proofs and identities.

Highlight: The page includes a proof of the identity (sin(x) + cos(x))² = 1 + 2sin(α)cos(α), demonstrating the interconnectedness of trigonometric functions.

Key points:

Corrections for previous calculations, emphasizing precision in trigonometric problem-solving
Advanced trigonometric identities and their proofs
Graphical representations of sine and cosine functions

Vocabulary: Trigonometric identity: An equation involving trigonometric functions that is true for all values of the variables involved.

The page concludes with graphical representations of sine and cosine functions, reinforcing the visual understanding of these trigonometric concepts and their periodic nature.

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Werde Teil der Community

Verbessere deine Noten

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Trigonometric Functions and the Unit Circle

This page delves into more advanced concepts related to trigonometric functions and their representation on the unit circle.

Definition: The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system, used to visualize trigonometric functions.

Key topics covered include:

Solving equations involving sine and cosine for specific angle ranges
Analyzing the behavior of trigonometric functions over extended angle ranges (0° to 720°)
Visualizing trigonometric functions on the unit circle and as graphs

Example: The page includes a problem asking for which angles between 0° and 720° the equation sin α = 0.75 is true, requiring students to consider multiple rotations around the unit circle.

The solutions demonstrate how to use the unit circle to find multiple angle solutions and how to interpret trigonometric functions graphically.

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Werde Teil der Community

Verbessere deine Noten

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Advanced Trigonometric Concepts

This page continues with solutions to more complex trigonometric problems, including those involving multiple triangles and advanced trigonometric identities.

Vocabulary: Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

The solutions cover:

Calculating distances and heights using trigonometric ratios
Finding areas of triangles using trigonometric functions
Applying trigonometric identities to solve equations

Highlight: One problem involves calculating the area of a triangle using the formula A = ½ * base * height, where the height is found using the sine function.

The page also includes partial solutions and teacher annotations, indicating areas where students may have made errors or where additional explanation might be needed.

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Werde Teil der Community

Verbessere deine Noten

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Solving Trigonometric Problems

This page demonstrates solutions to the problems presented in the test paper. It showcases various techniques for solving trigonometric equations and real-world applications.

Highlight: The solution to the tree problem shows how to use the sine function to calculate the tree's original height, which is approximately 8.66 meters.

The page includes detailed calculations for:

Using sin, cos, tan to calculate angles in a triangle
Applying the law of sines and cosines
Solving for unknown side lengths in complex triangles

Example: In one solution, the sine law is used to find an angle: sin α / a = sin β / b, where α and β are angles, and a and b are the opposite sides.

The solutions demonstrate the step-by-step process of solving trigonometric problems, emphasizing the importance of proper formula application and algebraic manipulation.