Asignaturas

Matemáticas

Trigonometría

Semejanza y trigonometría

Ejercicios Resueltos: Reducción al Primer Cuadrante y Sistema Sexagesimal para Primaria y ESO

Name: Knowunity
Brand: Knowunity

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Ejercicios Resueltos: Reducción al Primer Cuadrante y Sistema Sexagesimal para Primaria y ESO

Noe

@noe07

216 Seguidores

Seguir

The document provides a comprehensive guide on trigonometry concepts and exercises for students. It covers angle reduction to the first quadrant, the sexagesimal system, and trigonometric ratios.

Key points:
• Explains how to solve trigonometric problems involving sine, cosine, and tangent
• Covers conversion between sexagesimal and decimal angle measurements
• Demonstrates trigonometric ratios for common angles (0°, 30°, 45°, 60°, 90°)
• Includes solved example problems and practice exercises

28/4/2023

4291

ACT ^^
El cos de un ángulo del primer cuadrante es 3/4, calcula el seno del ángulo.
- ³/4
sen (+)
cas (+)
+g (+)
tg = 122
Sen x = d?
Isen +

Trigonometry Fundamentals and Problem Solving

This section introduces core trigonometric concepts and problem-solving techniques for angles in different quadrants.

Key topics covered:

Calculating sine and cosine values given other trigonometric ratios
Solving for unknown sides and angles in right triangles
Using trigonometric identities

Example: For an angle in the first quadrant with cosine 3/4, the sine is calculated as √(1 - cos^2) = √(1 - (3/4)^2) = √(7/16) = √7/4.

Highlight: The fundamental trigonometric identity sin^2 x + cos^2 x = 1 is used frequently to solve problems.

Vocabulary: Tangent (tg) - The ratio of sine to cosine for an angle

Ver

Trigonometry Problem Solving

This final section provides more complex problem-solving examples involving right triangles.

Key problem types:

Finding the hypotenuse given an angle and opposite side
Calculating cathetuses (legs) given the hypotenuse and an angle
Determining missing sides and angles in right triangles

Example: In a right triangle with hypotenuse 26 cm and an angle of 66°, the opposite cathetus is calculated as 26 × sin(66°) ≈ 23.74 cm.

Vocabulary: Cathetus (plural: cathetuses) - A leg of a right triangle; the side adjacent to the right angle

The problems demonstrate how to apply trigonometric ratios and the Pythagorean theorem to solve for unknown elements in right triangles, reinforcing the practical application of the concepts covered throughout the document.

Ver

Trigonometric Ratios for Any Angle

This section extends trigonometric concepts to angles in all quadrants and provides practice exercises.

Key topics:

Expressing angles in radians
Determining signs of trigonometric functions in different quadrants
Solving right triangle problems with various given information

Example: Express 240° in radians: 240° × (π rad / 180°) = 4π/3 rad

Highlight: The signs of trigonometric functions change depending on the quadrant of the angle.

Practice exercises include:

Converting between degrees and radians
Finding missing sides in right triangles given an angle and one side
Calculating unknown angles in right triangles

Ver

International System and Trigonometric Ratios

This section covers the conversion between degrees and radians, as well as the basic trigonometric ratios.

Key topics:

Converting between degrees and radians
Defining sine, cosine, tangent and their reciprocals
Trigonometric ratios in different quadrants

Vocabulary:

Radian - The angle subtended at the center of a circle by an arc equal in length to the radius

Cosecant (cosec) - The reciprocal of sine

Secant (sec) - The reciprocal of cosine

Cotangent (cotg) - The reciprocal of tangent

Highlight: The conversion factor between degrees and radians is π/180°.

The page also includes a diagram showing how trigonometric ratios relate to the sides of a right triangle in each quadrant.

Ver

Sexagesimal System and Angle Measurements

This section explains the sexagesimal system for measuring angles and how to convert between different angle representations.

Key concepts:

Incomplete and complete units in the sexagesimal system
Converting between degrees, minutes, and seconds
Adding and subtracting angles in sexagesimal form
Handling angles greater than 360°

Example: To convert 32.75° to the sexagesimal system: 32° + (0.75 × 60') = 32° 45'

Definition: Sexagesimal system - An angle measurement system based on 60, using degrees, minutes, and seconds.

The page also demonstrates how to perform arithmetic operations with complex angle measurements and deal with angles exceeding 360°.

Ver

Fundamental Trigonometric Relationships

This section presents the core trigonometric identities and demonstrates their proofs.

Key relationships covered:

sin^2 x + cos^2 x = 1
tan x = sin x / cos x
1 + tan^2 x = 1 / cos^2 x

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

Example: Proof of sin^2 x + cos^2 x = 1 using the Pythagorean theorem in a right triangle.

The page also includes a table of trigonometric ratios for common angles (0°, 30°, 45°, 60°, 90°), which is essential for solving many trigonometry problems.

Ver

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Knowunity es la app educativa nº 1 en cinco países europeos

Knowunity fue un artículo destacado por Apple y ha ocupado sistemáticamente los primeros puestos en las listas de la tienda de aplicaciones dentro de la categoría de educación en Alemania, Italia, Polonia, Suiza y Reino Unido. Regístrate hoy en Knowunity y ayuda a millones de estudiantes de todo el mundo.

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Knowunity es la app educativa nº 1 en cinco países europeos

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valoración media de la app

13 M

A los alumnos les encanta Knowunity

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Usuario de iOS

Me encanta esta app [...] ¡¡¡Recomiendo Knowunity a todo el mundo!!! Pasé de un 2 a un 9 con él :D

Javi, usuario de iOS

La app es muy fácil de usar y está muy bien diseñada. Hasta ahora he encontrado todo lo que estaba buscando y he podido aprender mucho de las presentaciones.

Mari, usuario de iOS

Me encanta esta app ❤️, de hecho la uso cada vez que estudio.

Asignaturas

Matemáticas

Trigonometría

Semejanza y trigonometría

Ejercicios Resueltos: Reducción al Primer Cuadrante y Sistema Sexagesimal para Primaria y ESO

Noe

@noe07

216 Seguidores

Seguir

The document provides a comprehensive guide on trigonometry concepts and exercises for students. It covers angle reduction to the first quadrant, the sexagesimal system, and trigonometric ratios.

28/4/2023

4291

4° ESO/1º Bach

Matemáticas

270

Trigonometry Fundamentals and Problem Solving

This section introduces core trigonometric concepts and problem-solving techniques for angles in different quadrants.

Key topics covered:

Calculating sine and cosine values given other trigonometric ratios
Solving for unknown sides and angles in right triangles
Using trigonometric identities

Example: For an angle in the first quadrant with cosine 3/4, the sine is calculated as √(1 - cos^2) = √(1 - (3/4)^2) = √(7/16) = √7/4.

Highlight: The fundamental trigonometric identity sin^2 x + cos^2 x = 1 is used frequently to solve problems.

Vocabulary: Tangent (tg) - The ratio of sine to cosine for an angle

Trigonometry Problem Solving

This final section provides more complex problem-solving examples involving right triangles.

Key problem types:

Finding the hypotenuse given an angle and opposite side
Calculating cathetuses (legs) given the hypotenuse and an angle
Determining missing sides and angles in right triangles

Example: In a right triangle with hypotenuse 26 cm and an angle of 66°, the opposite cathetus is calculated as 26 × sin(66°) ≈ 23.74 cm.

Vocabulary: Cathetus (plural: cathetuses) - A leg of a right triangle; the side adjacent to the right angle

Trigonometric Ratios for Any Angle

This section extends trigonometric concepts to angles in all quadrants and provides practice exercises.

Key topics:

Expressing angles in radians
Determining signs of trigonometric functions in different quadrants
Solving right triangle problems with various given information

Example: Express 240° in radians: 240° × (π rad / 180°) = 4π/3 rad

Highlight: The signs of trigonometric functions change depending on the quadrant of the angle.

Practice exercises include:

Converting between degrees and radians
Finding missing sides in right triangles given an angle and one side
Calculating unknown angles in right triangles

International System and Trigonometric Ratios

This section covers the conversion between degrees and radians, as well as the basic trigonometric ratios.

Key topics:

Converting between degrees and radians
Defining sine, cosine, tangent and their reciprocals
Trigonometric ratios in different quadrants

Vocabulary:

Radian - The angle subtended at the center of a circle by an arc equal in length to the radius

Cosecant (cosec) - The reciprocal of sine

Secant (sec) - The reciprocal of cosine

Cotangent (cotg) - The reciprocal of tangent

Highlight: The conversion factor between degrees and radians is π/180°.

The page also includes a diagram showing how trigonometric ratios relate to the sides of a right triangle in each quadrant.

Sexagesimal System and Angle Measurements

This section explains the sexagesimal system for measuring angles and how to convert between different angle representations.

Key concepts:

Incomplete and complete units in the sexagesimal system
Converting between degrees, minutes, and seconds
Adding and subtracting angles in sexagesimal form
Handling angles greater than 360°

Example: To convert 32.75° to the sexagesimal system: 32° + (0.75 × 60') = 32° 45'

Definition: Sexagesimal system - An angle measurement system based on 60, using degrees, minutes, and seconds.

The page also demonstrates how to perform arithmetic operations with complex angle measurements and deal with angles exceeding 360°.

Fundamental Trigonometric Relationships

This section presents the core trigonometric identities and demonstrates their proofs.

Key relationships covered:

sin^2 x + cos^2 x = 1
tan x = sin x / cos x
1 + tan^2 x = 1 / cos^2 x

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

Example: Proof of sin^2 x + cos^2 x = 1 using the Pythagorean theorem in a right triangle.

The page also includes a table of trigonometric ratios for common angles (0°, 30°, 45°, 60°, 90°), which is essential for solving many trigonometry problems.

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Knowunity es la app educativa nº 1 en cinco países europeos

Knowunity fue un artículo destacado por Apple y ha ocupado sistemáticamente los primeros puestos en las listas de la tienda de aplicaciones dentro de la categoría de educación en Alemania, Italia, Polonia, Suiza y Reino Unido. Regístrate hoy en Knowunity y ayuda a millones de estudiantes de todo el mundo.

Descargar en

Google Play

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4.9+

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en las listas de aplicaciones educativas de 12 países

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Usuario de iOS

Me encanta esta app [...] ¡¡¡Recomiendo Knowunity a todo el mundo!!! Pasé de un 2 a un 9 con él :D

Javi, usuario de iOS

La app es muy fácil de usar y está muy bien diseñada. Hasta ahora he encontrado todo lo que estaba buscando y he podido aprender mucho de las presentaciones.

Mari, usuario de iOS