This document provides an overview of trigonometry and its applications. It begins with definitions of trigonometry, its history and etymology. It discusses trigonometric functions like sine, cosine and their properties. It covers trigonometric identities and applications in fields like astronomy, navigation, acoustics and more. It also discusses angle measurement in degrees and radians. Laws of sines and cosines are explained. The document concludes with examples of trigonometric equations and their applications.
Trigonometry is the branch of mathematics that deals with triangles, especially right triangles. It has been used for over 4000 years, originally to calculate sundials. Key trigonometric functions are the sine, cosine, and tangent, which relate the angles and sides of a right triangle. Trigonometric identities and the trig functions of complementary angles are also discussed. Trigonometry has many applications, including in astronomy, navigation, engineering, optics, and more. It allows curved surfaces to be approximated in architecture using flat panels at angles.
Trigonometry is derived from Greek words meaning "three angles" and "measure". It deals with relationships between sides and angles of triangles, especially right triangles. The document discusses the history of trigonometry dating back to ancient Egypt and Babylon, and how it advanced through the works of Greek astronomer Hipparchus and Ptolemy. It also discusses the six trigonometric ratios and their formulas, various trigonometric identities, and applications of trigonometry in fields like architecture, engineering, astronomy, music, optics, and more.
This project on trigonometry was designed by two 10th grade students to introduce various topics in trigonometry. It includes sections on the introduction and definition of trigonometry, trigonometric ratios and their names in a right triangle, examples of applying ratios to find unknown sides, reciprocal identities of ratios, types of problems involving calculating ratios and evaluating expressions, value tables for common angles, formulas relating ratios, and main trigonometric identities. The project was created under the guidance of the students' mathematics teacher.
This document provides an overview of trigonometry presented by Vijay. It begins by listing the materials needed and encouraging note taking. The presentation then defines trigonometric ratios like sine, cosine and tangent using a right triangle. It also covers trigonometric ratios of specific angles like 45 and 30 degrees as well as complementary angles. The document concludes by explaining several trigonometric identities and providing a short summary of key points.
This document provides an introduction to trigonometry. It defines trigonometry as dealing with relations of sides and angles of triangles. It discusses the history of trigonometry and defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent). It provides the ratios for some specific angles and identities relating the ratios. It describes applications of trigonometry in fields like astronomy, navigation, architecture, and more.
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It emerged during the 3rd century BC from applications of geometry to astronomy. Hipparchus is considered the founder of trigonometry, compiling the first trigonometric table in the 2nd century BC. Key trigonometric functions like sine, cosine, and tangent were discovered between the 5th-10th centuries CE by mathematicians including Aryabhata, Muhammad ibn Musa al-Khwarizmi, and Abu al-Wafa. Trigonometry is applied to calculate angles of elevation and depression used in applications like determining the angle an airplane is viewed from the ground.
Trigonometry is the branch of mathematics that deals with triangles, especially right triangles. It has been used for over 4000 years, originally to calculate sundials. Key trigonometric functions are the sine, cosine, and tangent, which relate the angles and sides of a right triangle. Trigonometric identities and the trig functions of complementary angles are also discussed. Trigonometry has many applications, including in astronomy, navigation, engineering, optics, and more. It allows curved surfaces to be approximated in architecture using flat panels at angles.
Trigonometry is derived from Greek words meaning "three angles" and "measure". It deals with relationships between sides and angles of triangles, especially right triangles. The document discusses the history of trigonometry dating back to ancient Egypt and Babylon, and how it advanced through the works of Greek astronomer Hipparchus and Ptolemy. It also discusses the six trigonometric ratios and their formulas, various trigonometric identities, and applications of trigonometry in fields like architecture, engineering, astronomy, music, optics, and more.
This project on trigonometry was designed by two 10th grade students to introduce various topics in trigonometry. It includes sections on the introduction and definition of trigonometry, trigonometric ratios and their names in a right triangle, examples of applying ratios to find unknown sides, reciprocal identities of ratios, types of problems involving calculating ratios and evaluating expressions, value tables for common angles, formulas relating ratios, and main trigonometric identities. The project was created under the guidance of the students' mathematics teacher.
This document provides an overview of trigonometry presented by Vijay. It begins by listing the materials needed and encouraging note taking. The presentation then defines trigonometric ratios like sine, cosine and tangent using a right triangle. It also covers trigonometric ratios of specific angles like 45 and 30 degrees as well as complementary angles. The document concludes by explaining several trigonometric identities and providing a short summary of key points.
This document provides an introduction to trigonometry. It defines trigonometry as dealing with relations of sides and angles of triangles. It discusses the history of trigonometry and defines the six trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent). It provides the ratios for some specific angles and identities relating the ratios. It describes applications of trigonometry in fields like astronomy, navigation, architecture, and more.
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It emerged during the 3rd century BC from applications of geometry to astronomy. Hipparchus is considered the founder of trigonometry, compiling the first trigonometric table in the 2nd century BC. Key trigonometric functions like sine, cosine, and tangent were discovered between the 5th-10th centuries CE by mathematicians including Aryabhata, Muhammad ibn Musa al-Khwarizmi, and Abu al-Wafa. Trigonometry is applied to calculate angles of elevation and depression used in applications like determining the angle an airplane is viewed from the ground.
Trigonometry deals with triangles and the angles between sides. The main trigonometric ratios are defined using the sides of a right triangle: sine, cosine, and tangent. Trigonometric functions can convert between degrees and radians. Standard angle positions and trigonometric identities relate trig functions of summed and subtracted angles. The sine and cosine rules relate the sides and angles of any triangle, allowing for calculations of missing sides or angles given other information. Unit circle graphs further illustrate trigonometric functions.
Trigonometry studies triangles and relationships between sides and angles. This document discusses using trigonometric ratios to calculate heights and distances, including the angles of elevation and depression. It provides examples of using trigonometry to find the height of a tower from the angle of elevation measured 30 meters away (30 meters high), and the height of a pole from the angle made by a rope tied to its top (10 meters high). It also explains calculating the length of a kite string from the angle of elevation.
Trigonometry is the study of triangles and their relationships. The document discusses how trigonometry is used in fields like architecture, astronomy, geology, and for measuring distances and heights. It provides examples of how trigonometry can be used to calculate the height of a building given the distance and angle of elevation to its top.
Trigonometry is used to calculate unknown heights, distances, and angles using relationships between sides and angles of triangles. It was developed by ancient Greek mathematicians like Thales and Hipparchus to solve problems in astronomy and geography. Some key applications include using trigonometric ratios like tangent and cotangent along with known distances and angles of elevation/depression to determine the height of objects like towers, buildings, and mountains when direct measurement is not possible. The document provides historical context and examples to illustrate how trigonometric concepts have been applied to problems involving finding heights, distances, and other unknown measurements through the use of triangles and their properties.
Trigonometry is the study of relationships between the sides and angles of triangles. It has its origins over 4000 years ago in ancient Egypt, Mesopotamia, and the Indus Valley. The first recorded use was by the Greek mathematician Hipparchus around 150 BC. Trigonometry defines trigonometric functions like sine, cosine, and tangent that relate angles and sides of a triangle. It has many applications in fields like astronomy, navigation, engineering, and more.
Kartikeya Pandey thanks his teacher Ms. Meha Bhargava and principal Ms. Jasleen Kaur for allowing him to complete a project on trigonometry. He also thanks his parents and friends for their help. The document then provides information on trigonometry including its origins in ancient mathematics, definitions of key terms like sine, cosine, and tangent. It also discusses right triangles, angle measurement in degrees and radians, trigonometric functions, trigonometric identities, and applications of trigonometry.
Maths ppt on some applications of trignometryHarsh Mahajan
This document discusses trigonometry and how it can be used to calculate heights and distances. It defines trigonometric ratios and the angles of elevation and depression. It then provides examples of using trigonometry to calculate the height of a tower given the angle of elevation is 30 degrees and the distance from the observer is 30 meters. It also gives an example of calculating the height of a pole using the angle made by the rope and the ground.
This document provides an overview of trigonometry, including its origins in Greek mathematics, the six main trigonometric functions defined in terms of right triangles, and trigonometric identities. Trigonometry is the study of triangles and relationships between sides and angles, with the six functions—sine, cosine, tangent, cotangent, secant, and cosecant—defined based on ratios of sides. Special angle values and identities are also discussed as important concepts in trigonometry.
This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors
Trigonometry deals with relationships between sides and angles of triangles. It originated in ancient Greece and was used to calculate sundials. Key concepts include trigonometric functions like sine, cosine and tangent that relate a triangle's angles to its sides. Trigonometric identities and angle formulae allow for the conversion between functions. It has wide applications in fields like astronomy, engineering and navigation.
Trigonometry is the branch of mathematics dealing with relationships between sides and angles of triangles. The document traces the origins of common trigonometric functions like sine, cosine, and tangent from ancient Indian and Greek mathematicians. It then explains how trigonometric ratios are used to calculate the angle of elevation of a room by measuring the wall length and diagonal distance and applying the sine ratio formula.
Trigonometry studies triangles and relationships between sides and angles. This document discusses using trigonometric ratios to calculate heights and distances, such as of trees, towers, water tanks, and distance from a ship to a lighthouse. It provides examples of using trigonometry to calculate the height of a tower given the angle of elevation and distance from its base, and the height of a pole given the angle made by the rope tied to its top and the ground.
Trigonometry deals with relationships between sides and angles of triangles, especially right triangles. It has been studied since ancient times and is used across many fields including astronomy, navigation, architecture, engineering, and digital imaging. Trigonometric functions relate ratios of sides of a right triangle to an angle of the triangle. These functions and their relationships are important tools that are applied in problems involving waves, forces, rotations, and more.
Trigonometry was invented by ancient Greeks to calculate distances and angles in astronomy. It relates the angles and lengths of triangles, allowing unknown values to be determined if one angle and one length are known. Some early applications included creating trigonometric tables for astronomical computations and using triangulation to measure heights and distances. Modern uses include navigation, surveying, engineering, acoustics, and technologies like GPS which rely on trigonometric calculations.
Lesson plan on introduction of trigonometry, students must aware about the history , concepts to be done, what common error they commit and what are the scope of this topic in careers
Trigonometry is the branch of mathematics that deals with relationships between the sides and angles of triangles, especially right triangles. It has been studied since ancient times by civilizations like Egypt, Mesopotamia, and India. Key concepts in trigonometry include trigonometric functions like sine, cosine, and tangent that relate ratios of sides of a right triangle to an angle of the triangle. Trigonometry has many applications in fields like astronomy, navigation, engineering, and more.
This document discusses trigonometric ratios and identities. It defines trigonometric ratios as relationships between sides and angles of a right triangle. Specific ratios are defined for angles of 0, 30, 45, 60, and 90 degrees. Complementary angle identities are examined, showing ratios are equal for complementary angles (e.g. sin(90-A)=cos(A)). Trigonometric identities are derived from the Pythagorean theorem, including cos^2(A) + sin^2(A) = 1, sec^2(A) = 1 + tan^2(A), and cot^2(A) + 1 = cosec^2(A). Examples are provided to demonstrate using identities when
The document discusses the history of trigonometry and the group's project to use trigonometry to measure the height of a church. It provides background on important historical figures who developed trigonometry like Hipparchus, Ptolemy, and Muslim astronomers. It then describes how the group used a homemade quadrant and trigonometric calculations to determine the church was approximately 14.57 meters tall. The group felt the project was a good learning experience but took a long time to complete carefully.
Trigonometry is a branch of mathematics that studies triangles and their relationships. The document discusses how trigonometry is used in the fields of architecture, astronomy, and geology. In architecture, trigonometry is used to calculate angles to ensure structural stability and safety. Astronomers use trigonometry and concepts like parallax to calculate distances between stars. Geologists use trigonometry to estimate true dip angles of bedding to determine slope stability important for building foundations.
Trigonometry is used in many fields including astronomy, architecture, navigation, chemistry, meteorology, engineering, carpentry, biology, and forensics. It allows measurement of distances to stars, calculation of angles and forces in building design, navigation on land and sea, modeling molecular structures, tracking weather balloons, structural design, angled cuts in carpentry, determining molecular structures through x-ray crystallography, and analyzing crime scenes. Trigonometric functions like sine, cosine, and tangent are essential mathematical tools across diverse applications in science and technology.
Trigonometry deals with right triangles and angles, and is used in fields like sound, light, and perceptions of beauty. The document defines common trigonometric terms like angle of elevation, angle of depression, sine, cosine, and tangent. It provides examples of using trigonometric functions to solve problems like finding the height of a tower or flagpole given angle of elevation measurements. Several practice problems on trigonometry are also presented and solved.
Trigonometry deals with triangles and the angles between sides. The main trigonometric ratios are defined using the sides of a right triangle: sine, cosine, and tangent. Trigonometric functions can convert between degrees and radians. Standard angle positions and trigonometric identities relate trig functions of summed and subtracted angles. The sine and cosine rules relate the sides and angles of any triangle, allowing for calculations of missing sides or angles given other information. Unit circle graphs further illustrate trigonometric functions.
Trigonometry studies triangles and relationships between sides and angles. This document discusses using trigonometric ratios to calculate heights and distances, including the angles of elevation and depression. It provides examples of using trigonometry to find the height of a tower from the angle of elevation measured 30 meters away (30 meters high), and the height of a pole from the angle made by a rope tied to its top (10 meters high). It also explains calculating the length of a kite string from the angle of elevation.
Trigonometry is the study of triangles and their relationships. The document discusses how trigonometry is used in fields like architecture, astronomy, geology, and for measuring distances and heights. It provides examples of how trigonometry can be used to calculate the height of a building given the distance and angle of elevation to its top.
Trigonometry is used to calculate unknown heights, distances, and angles using relationships between sides and angles of triangles. It was developed by ancient Greek mathematicians like Thales and Hipparchus to solve problems in astronomy and geography. Some key applications include using trigonometric ratios like tangent and cotangent along with known distances and angles of elevation/depression to determine the height of objects like towers, buildings, and mountains when direct measurement is not possible. The document provides historical context and examples to illustrate how trigonometric concepts have been applied to problems involving finding heights, distances, and other unknown measurements through the use of triangles and their properties.
Trigonometry is the study of relationships between the sides and angles of triangles. It has its origins over 4000 years ago in ancient Egypt, Mesopotamia, and the Indus Valley. The first recorded use was by the Greek mathematician Hipparchus around 150 BC. Trigonometry defines trigonometric functions like sine, cosine, and tangent that relate angles and sides of a triangle. It has many applications in fields like astronomy, navigation, engineering, and more.
Kartikeya Pandey thanks his teacher Ms. Meha Bhargava and principal Ms. Jasleen Kaur for allowing him to complete a project on trigonometry. He also thanks his parents and friends for their help. The document then provides information on trigonometry including its origins in ancient mathematics, definitions of key terms like sine, cosine, and tangent. It also discusses right triangles, angle measurement in degrees and radians, trigonometric functions, trigonometric identities, and applications of trigonometry.
Maths ppt on some applications of trignometryHarsh Mahajan
This document discusses trigonometry and how it can be used to calculate heights and distances. It defines trigonometric ratios and the angles of elevation and depression. It then provides examples of using trigonometry to calculate the height of a tower given the angle of elevation is 30 degrees and the distance from the observer is 30 meters. It also gives an example of calculating the height of a pole using the angle made by the rope and the ground.
This document provides an overview of trigonometry, including its origins in Greek mathematics, the six main trigonometric functions defined in terms of right triangles, and trigonometric identities. Trigonometry is the study of triangles and relationships between sides and angles, with the six functions—sine, cosine, tangent, cotangent, secant, and cosecant—defined based on ratios of sides. Special angle values and identities are also discussed as important concepts in trigonometry.
This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors
Trigonometry deals with relationships between sides and angles of triangles. It originated in ancient Greece and was used to calculate sundials. Key concepts include trigonometric functions like sine, cosine and tangent that relate a triangle's angles to its sides. Trigonometric identities and angle formulae allow for the conversion between functions. It has wide applications in fields like astronomy, engineering and navigation.
Trigonometry is the branch of mathematics dealing with relationships between sides and angles of triangles. The document traces the origins of common trigonometric functions like sine, cosine, and tangent from ancient Indian and Greek mathematicians. It then explains how trigonometric ratios are used to calculate the angle of elevation of a room by measuring the wall length and diagonal distance and applying the sine ratio formula.
Trigonometry studies triangles and relationships between sides and angles. This document discusses using trigonometric ratios to calculate heights and distances, such as of trees, towers, water tanks, and distance from a ship to a lighthouse. It provides examples of using trigonometry to calculate the height of a tower given the angle of elevation and distance from its base, and the height of a pole given the angle made by the rope tied to its top and the ground.
Trigonometry deals with relationships between sides and angles of triangles, especially right triangles. It has been studied since ancient times and is used across many fields including astronomy, navigation, architecture, engineering, and digital imaging. Trigonometric functions relate ratios of sides of a right triangle to an angle of the triangle. These functions and their relationships are important tools that are applied in problems involving waves, forces, rotations, and more.
Trigonometry was invented by ancient Greeks to calculate distances and angles in astronomy. It relates the angles and lengths of triangles, allowing unknown values to be determined if one angle and one length are known. Some early applications included creating trigonometric tables for astronomical computations and using triangulation to measure heights and distances. Modern uses include navigation, surveying, engineering, acoustics, and technologies like GPS which rely on trigonometric calculations.
Lesson plan on introduction of trigonometry, students must aware about the history , concepts to be done, what common error they commit and what are the scope of this topic in careers
Trigonometry is the branch of mathematics that deals with relationships between the sides and angles of triangles, especially right triangles. It has been studied since ancient times by civilizations like Egypt, Mesopotamia, and India. Key concepts in trigonometry include trigonometric functions like sine, cosine, and tangent that relate ratios of sides of a right triangle to an angle of the triangle. Trigonometry has many applications in fields like astronomy, navigation, engineering, and more.
This document discusses trigonometric ratios and identities. It defines trigonometric ratios as relationships between sides and angles of a right triangle. Specific ratios are defined for angles of 0, 30, 45, 60, and 90 degrees. Complementary angle identities are examined, showing ratios are equal for complementary angles (e.g. sin(90-A)=cos(A)). Trigonometric identities are derived from the Pythagorean theorem, including cos^2(A) + sin^2(A) = 1, sec^2(A) = 1 + tan^2(A), and cot^2(A) + 1 = cosec^2(A). Examples are provided to demonstrate using identities when
The document discusses the history of trigonometry and the group's project to use trigonometry to measure the height of a church. It provides background on important historical figures who developed trigonometry like Hipparchus, Ptolemy, and Muslim astronomers. It then describes how the group used a homemade quadrant and trigonometric calculations to determine the church was approximately 14.57 meters tall. The group felt the project was a good learning experience but took a long time to complete carefully.
Trigonometry is a branch of mathematics that studies triangles and their relationships. The document discusses how trigonometry is used in the fields of architecture, astronomy, and geology. In architecture, trigonometry is used to calculate angles to ensure structural stability and safety. Astronomers use trigonometry and concepts like parallax to calculate distances between stars. Geologists use trigonometry to estimate true dip angles of bedding to determine slope stability important for building foundations.
Trigonometry is used in many fields including astronomy, architecture, navigation, chemistry, meteorology, engineering, carpentry, biology, and forensics. It allows measurement of distances to stars, calculation of angles and forces in building design, navigation on land and sea, modeling molecular structures, tracking weather balloons, structural design, angled cuts in carpentry, determining molecular structures through x-ray crystallography, and analyzing crime scenes. Trigonometric functions like sine, cosine, and tangent are essential mathematical tools across diverse applications in science and technology.
Trigonometry deals with right triangles and angles, and is used in fields like sound, light, and perceptions of beauty. The document defines common trigonometric terms like angle of elevation, angle of depression, sine, cosine, and tangent. It provides examples of using trigonometric functions to solve problems like finding the height of a tower or flagpole given angle of elevation measurements. Several practice problems on trigonometry are also presented and solved.
Trigonometry is the branch of mathematics dealing with triangles and trigonometric functions of angles. It is derived from Greek words meaning "three angles" and "measure". Trigonometry specifically studies relationships between sides and angles of triangles, and calculations based on trigonometric functions like sine, cosine, and tangent. Trigonometry has many applications in fields like astronomy, navigation, architecture, engineering, and more.
This document discusses the correlation of mathematics with various domains:
1) Mathematics is correlated with life activities through concepts like percentages, interest rates, and ratios that are useful in everyday life.
2) Different branches of mathematics like arithmetic, algebra, geometry are interrelated through concepts like functions and mathematical structures.
3) Topics within the same branch of mathematics are also correlated, for example concepts in algebra relate to equations, and areas of shapes relate in geometry.
4) Mathematics is also correlated with other subjects like physical sciences through expression of laws as mathematical equations, with biology through use of higher math methods, and with engineering as mathematics forms the basis of engineering courses.
This document contains notes from a trigonometry class taught by Steven Butler at Brigham Young University in Fall 2002. It is divided into 9 chapters that cover topics such as geometric foundations, the Pythagorean theorem, angle measurement, trigonometry with right triangles, trigonometry with circles, graphing trigonometric functions, inverse trigonometric functions, and working with trigonometric identities. Each chapter contains sections that explain key concepts and include supplemental practice problems.
Trigonometry deals with relationships between sides and angles of triangles, especially right triangles. It has many applications in fields like architecture, astronomy, engineering, and more. The document provides background on trigonometry, defines trigonometric functions and ratios, discusses right triangles, and gives several examples of how trigonometry is used in areas like navigation, construction, and digital imaging.
APPLICATION OF LINEAR ALGEBRA IN ECONOMICSAmit Garg
This document discusses applications of linear algebra in economics, specifically the Leontiff Input-Output Model. The model shows interdependencies between sectors of the economy using linear equations. It represents the economy as a consumption matrix that shows quantities of inputs needed to produce one unit of output for each sector. Total production, internal demand, and final demand can also be represented as vectors and related through a linear equation that the model solves to determine production levels for each sector.
Correlation of subjects in school (b.ed notes)Namrata Saxena
This document discusses the concept of correlation in education. It defines correlation as the mutual relationship between different subjects or variables in a curriculum. The document outlines the importance of correlation, including that it helps students perceive knowledge as a whole, strengthens retention of knowledge, and promotes well-rounded development. It discusses different types of correlation, including vertical/internal correlation between topics within a subject and horizontal/external correlation between different subjects. Examples are provided of how mathematics can be correlated with other subjects like science, geography, and economics.
This document discusses various applications of linear algebra in different fields such as abstract thinking, chemistry, coding theory, cryptography, economics, elimination theory, games, genetics, geometry, graph theory, heat distribution, image compression, linear programming, Markov chains, networking, and sociology. It provides examples of how linear algebra concepts such as systems of linear equations and matrix operations are used in topics like balancing chemical equations, error detection in coding, encryption/decryption, economic models, genetic inheritance, and finding lines and circles in geometry.
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles in triangles. The key concepts are the trigonometric functions sine, cosine, and tangent, which describe ratios of sides of a right triangle. Trigonometry has applications in fields like navigation, music, engineering, and more. It has evolved significantly from its origins in ancient Greece and India, with modern definitions extending it to all real and complex number arguments.
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It emerged from applications of geometry to astronomy in the 3rd century BC. Trigonometric functions relate ratios of sides of right triangles to angles and allow for determination of all angles and sides from just one angle and one side. Trigonometry is used in many fields including astronomy, navigation, music, acoustics, optics, engineering, and more due to applications of triangulation and modeling periodic functions.
Trigonometry is the branch of mathematics dealing with relationships between the sides and angles of triangles. It has been used for over 4000 years, originally to calculate sundials and now in fields like navigation, engineering, and astronomy. Trigonometry specifically studies right triangles, where one angle is 90 degrees. The Pythagorean theorem relates the sides of a right triangle, and trigonometric ratios like sine, cosine, and tangent are used to calculate unknown sides and angles based on known values. Trigonometry has many applications in areas involving waves, geometry, and modeling real-world phenomena.
Trigonometry is the branch of mathematics dealing with relationships between the sides and angles of triangles. It has been used for over 4000 years, originally to calculate sundials and now in fields like astronomy, engineering, and digital imaging. Trigonometry specifically studies right triangles and defines trigonometric functions like sine, cosine, and tangent that relate a triangle's angles and sides. Key concepts include trigonometric ratios, the Pythagorean theorem, trigonometric identities, and applications to problems involving distance, direction, and waves.
Trigonometry is the branch of mathematics dealing with relationships between the sides and angles of triangles. It has been developed and used for over 4000 years, originating in ancient civilizations for purposes like calculating sundials. A key foundation is the right triangle, where one angle is 90 degrees. Pythagoras' theorem relates the sides of a right triangle, and trigonometric ratios like sine, cosine, and tangent define relationships between sides and angles. Trigonometry has many applications, from astronomy and navigation to engineering, physics, and digital imaging.
Trigonometry is the branch of mathematics that studies relationships involving lengths and angles of triangles. Key concepts include:
- The trigonometric functions (sine, cosine, tangent, etc.) relate angles and side lengths of triangles and are most simply defined using the unit circle.
- Trigonometry has its roots in ancient Greek, Indian, Chinese, Islamic and European mathematics. Important early contributors include Hipparchus, Ptolemy and Aryabhata.
- Right-angled triangle definitions establish the hypotenuse, opposite and adjacent sides and define the trigonometric functions in terms of ratios of these sides.
Trigonometry is the branch of mathematics dealing with relationships between the sides and angles of triangles and the calculations of trigonometric functions. It has many applications in fields like navigation, surveying, physics, engineering and more. Some key aspects covered are the definitions of trigonometric functions like sine, cosine and tangent; common trigonometric identities; formulae for calculating unknown sides and angles of triangles; and the history of trigonometry dating back to ancient Greek and Indian mathematicians.
Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged from applications of geometry to astronomy in the 3rd century BC. Trigonometric functions relate ratios of sides of a right triangle to its angles, and are now used across many fields including physics, engineering, music, astronomy, and more. Key concepts include defining the sine, cosine, and tangent functions; extending them to angles beyond 90 degrees using the unit circle; and common trigonometric identities and formulas used for solving triangles.
Trigonometry is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. It defines trigonometric functions that describe these relationships and are applicable to cyclical phenomena like waves. Trigonometry has its origins in ancient Greek and Indian mathematics and was further developed by Islamic mathematicians. It is the foundation of surveying and has many applications in fields like astronomy, music, acoustics, and more.
Trigonometry is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. It defines trigonometric functions that describe these relationships and are applicable to cyclical phenomena like waves. Trigonometry has its origins in ancient Greek and Indian mathematics and was further developed by Islamic mathematicians. It is the foundation of surveying and has many applications in fields like astronomy, music, acoustics, and more.
Trigonometry is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. It defines trigonometric functions that describe these relationships and are applicable to cyclical phenomena like waves. Trigonometry has its origins in ancient Greek and Indian mathematics and was further developed by Islamic mathematicians. It is the foundation of surveying and has many applications in fields like astronomy, music, acoustics, and more.
The document discusses trigonometry and provides definitions and examples of its uses. It defines trigonometry as the branch of mathematics dealing with relationships involving lengths and angles of triangles. It then gives examples of how trigonometry is used in fields like navigation, architecture, engineering, and game development. It also provides information on trigonometric functions like sine waves and their importance in fields like physics and signal processing.
Trigonometry is a branch of mathematics that deals with relationships between the sides and angles of triangles, especially right triangles. It has many applications in fields like astronomy, navigation, engineering, and more. Some key uses of trigonometry include measuring inaccessible heights and distances by using trigonometric functions and properties of triangles formed by angles of elevation or depression. For example, trigonometry can be used to calculate the height of a building or tree by measuring the angle of elevation from a known distance away. It also has applications in measuring distances in astronomy, designing curved architectural structures, and calculating road grades. The document provides examples of various real-world applications of trigonometric concepts.
this is a slide share on introduction of trigonometry this slide share includes every single information about the lesson trigonometry and this is best for class 10
Trigonometry is the study of measuring triangles and angles. It originated over 4000 years ago in ancient Egypt, Mesopotamia, and India to calculate sundials and solve triangles. Key developments include Hipparchus' trig tables in 150 BC and the Sulba Sutras in 800-500 BC. Trigonometry has many applications including astronomy, navigation, music, acoustics, optics, engineering, and more due to its ability to model waves and approximate curved surfaces with triangles. It remains an important area of ongoing research.
Trigonometry is the branch of mathematics that deals with relationships between sides and angles of triangles, especially right triangles. It has its origins in ancient civilizations over 4000 years ago and was originally used to calculate sundials. Key concepts in trigonometry include the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) that relate ratios of sides and angles in triangles. Trigonometric functions have many applications in fields like astronomy, navigation, engineering, and more.
Trigonometry deals with relationships between sides and angles of triangles, especially right triangles. It has been used for thousands of years in fields like astronomy, navigation, architecture, engineering, and more modern fields like digital imaging and computer graphics. Trigonometric functions define ratios between sides of a right triangle and are used to solve for unknown sides and angles. Common applications include calculating distances, heights, satellite positioning, and modeling waves and vibrations.
Trigonometry deals with relationships between sides and angles of triangles, especially right triangles. It has been studied since ancient civilizations over 4000 years ago and is used in many fields today including architecture, astronomy, engineering, and more. Trigonometric functions relate ratios of sides in a right triangle to the angles of the triangle. These functions and their relationships are important tools that allow calculations and problem solving across various domains.
The document discusses trigonometry, which studies triangles and relationships between lengths of sides and angles. It defines the trigonometric functions of sine, cosine, and tangent, which describe these relationships. Trigonometry has applications in fields like surveying, physics, and engineering. The history of trigonometry is then summarized, noting contributions from ancient Greek, Indian, Islamic, and Chinese mathematicians.
Trigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations. It emerged in the 3rd century BC from applications of geometry to astronomy. There are six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - that are used to obtain unknown angles and distances from known or measured angles in geometric figures. Trigonometry studies relationships involving lengths and angles of triangles and is commonly used to find distances and heights.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
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The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
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significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
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'Land uses,' which are determined by both human activities and the physical characteristics of the
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The utilization of land is impacted by human needs and environmental factors. In countries
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to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
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Accurate understanding of land use and cover is imperative for the development planning
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changes, conversion trends, and other related patterns. The spatial dimensions of land use and
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these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
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Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
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How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
3. “Indeed we have created everything
in a proper measure.”
(Surah Al-Qamr)
Allah says in the Holy Quran
4. Trigonometry and Applications
Fields of discussion
What is Mathematics?
Prince of Mathematicians
What is Trigonometry?
History and the meaning of the word sine and cosine.
Trigonometric functions, Circular functions or cyclometric functions
Fields of Trigonometry
Ancient Egypt and the Mediterranean world
Applications of Trigonometry
Angle measurement
Properties of sines and cosines
The Law(Rule) of sines,cosines
Trigonometric Equations
Applications of Trigonometric Equations
5. What is Mathematics?
Etymology
The word “Mathematics" comes from the Greek word (máthēma), which
means learning, study, science, and additionally more technical
meaning “Mathematical study",
Mathematics (Definition)
A group of related subjects , including ALGEBRA, GEOMETRY,
TRIGONOMETRY and CALCULUS, concerned with the study of number
,quantity, structure, shape and space.
Applications
Mathematics is used throughout the world as an essential tool in many
fields, including natural science, engineering, medicine, and the social
sciences.
6. Prince of Mathematicians
Carl Friedrich Gauss
Himself known as the
"prince of mathematicians“,
referred to Mathematics as
"the Queen of the Sciences".
This German Mathematician contributed
to many areas of Mathematics, including
probability theory, algebra, and geometry.
He proved that every polynomial has at
least one root, or solution; this theory is
known as the fundamental theory
of algebra. Gauss also applied his
mathematical work to theories of
electricity and magnetism.
7. What is Trigonometry?
Etymology
The word Trigonometry is derived from three Greek words ‘tries’(three),
‘goni’(angle) and ‘metron’(measurement). So literally, this word means
“measurement of the triangle”.
Trigonometry (Definition)
The branch of Mathematics concerned with the properties of
trigonometric functions and their application to the determination of the
sides and angles of triangles.
Trigonometry has now a wide application in higher Mathematics
in fact, any attempt to study Higher Mathematics would be an utter
failure without a working knowledge of trigonometry.
It has applications in both pure mathematics and applied mathematics,
where it is essential in many branches of science and technology.
8. History and the meaning of the word sine and cosine
Interesting word history for "sine”
The Hindu mathematician Aryabhata (about 475–550 A.D.) used the
Sanskrit word “jya” or “jiva” for the half-chord which was sometimes
shortened to jiva. This was brought into Arabic as jiba, and written in Arabic
simply with two consonants jb, vowels not being written. Later, Latin
translators selected the word sinus to translate jb thinking that the word was
an arabic word jaib, which meant bosom, fold, or bay, The Latin word for
bosom, bay, or curve is “sinus”. In English, sinus was imported as "sine".
This word history for "sine" is interesting because it follows the path of
trigonometry from India, through the Arabic language from Baghdad
through Spain, into western Europe in the Latin language, and then to
modern languages such as English.
9. Trigonometric functions, Circular functions or
cyclometric functions
Any of a group of functions expressible in terms of the ratios of
the sides of right-angled triangle.
Sine Ratio The sine of an angle in a right triangle equals the opposite
side divided by the hypotenuse:
sin =opp/hyp
Cosine Ratio. Cosines are just sines of the complementary angle. Thus,
the name "cosine" ("co" being the first two letters of "complement").
The complementary angle equals the given angle subtracted from a right
angle, 90°. For instance, if the angle is 30°, then its complement is 60°.
Generally, for any angle x,
cos =adj/hyp
cos x = sin (90° – x). Or cos 50 = sin (90 – 50) = sin40ᵒ ᵒ
Tangent Ratio tanx = sinx/cosx
tanx = opp/adj
10. Trigonometric functions, Circular functions or
cyclometric functions
Secant:
sec q = 1/cos q
Cosecant:
csc q = 1/sin q
Cotangent:
cot q = 1/ tan q
cot q = cos q/sin q
tan q = sin q/cos q
12. Trigonometric Identities
The following formulas, called identities, which show the relationships
between the trigonometric functions, hold for all values of the angle θ, or of
two angles, θ and φ, for which the functions involved are
13.
14. Fields of Trigonometry
Plane Trigonometry
In many applications of trigonometry the
essential problem is the solution of triangles.
If enough sides and angles are known, the
remaining sides and angles as well as
the area can be calculated, and the triangle
is then said to be solved. Triangles can be
solved by the law of sines and the law of
cosines.
Surveyors apply the principles of geometry
and trigonometry in determining the shapes,
measurements and position of features on
or beneath the surface of the Earth. Such
topographic surveys are useful in the design
of roads, tunnels, dams, and other structures.
15. Fields of Trigonometry
Spherical Trigonometry
Spherical Trigonometry involves the study of spherical triangles,
which are formed by the intersection of three great circle arcs on the
surface of a sphere.
Great Circle
A great circle is a theoretical circle,
such as the equator, formed by the
intersection of the earth’s surface
and an imaginary plane that passes
through the center of the earth
and divides it into two equal parts.
Navigators use great circles to find
the shortest distance between any
Two points on the earth’s surface.
16. Fields of Trigonometry
Analytic Trigonometry
Analytic Trigonometry combines the use of a coordinate system,
such as the Cartesian coordinate system used in analytic
geometry, with algebraic manipulation of the various
trigonometry functions to obtain formulas useful for scientific
and engineering applications.
17.
18. Ancient Egypt and the Mediterranean world
Several ancient civilizations—in particular, the Egyptian,
Babylonian, Hindu, and Chinese—possessed a considerable
knowledge of practical geometry, including some concepts of
trigonometry.
A close analysis of the text, with its accompanying figures,
reveals that this word means the slope of an incline, essential
knowledge for huge construction projects such as the pyramids.
It shows that the Egyptians had at least some knowledge of the
numerical relations in a triangle, a kind of “proto-
trigonometry.”
20. Applications of Trigonometry
.
Fields that use trigonometry or trigonometric functions include Astronomy
(especially for locating apparent positions of celestial objects(star or planet),
in which spherical trigonometry is essential) and hence navigation (on the
oceans, in aircraft, and in space), to measure distances between landmarks,
and in satellite navigation systems. The sine and cosine functions are
fundamental to the theory of periodic functions such as those that describe
sound and light waves.
Music theory, acoustics(study of sound ), optics, electronics, probability
theory, statistics, biology, medical imaging (CAT scans and ultrasound),
pharmacy, chemistry, number theory cryptology(coding), seismology,
meteorology, oceanography, many physical sciences, land surveying and
geodesy(cartography), architecture, phonetics (sounds of human speech),
economics, electrical engineering, mechanical engineering, civil engineering,
computer graphics, crystallography and game development.
21. Applications of Trigonometry
Marine sextants like this are used to measure the angle of the sun or
stars with respect to the horizon. Using trigonometry and a marine
chronometer(timer), the position of the ship can then be determined from
several such measurements.
22. Applications of Trigonometry
Wave Mathematics
Waves are familiar to us from the ocean, the study of sound, earthquakes, and
other natural phenomenon. Ocean waves come in very different sizes to fully
understand waves, we need to understand measurements associated with
these waves, such as how often they repeat (their frequency), and how long
they are (their wavelength), and their vertical size (amplitude).
The importance of the sine and
cosine functions is in describing
periodic phenomena—the
vibrations of a violin string,
the oscillations of a clock pendulum,
or the propagation of electromagnetic
waves, sound and light waves.
23. Applications of Trigonometry
Sine waves in nature
i)Sound waves are sine waves whenever we listen to music ,
we are actually listening to sound waves.
ii) light waves are also sine waves.
iii)Radio waves are sine waves.
iv)Simple harmonic motion of a spring when pulled
and released is a sine wave.
v) Alternating current (AC) is a sine wave.
vi) Pendulum clock oscillations are sinusoidal in
nature
vii) Waves of ocean are sinusoidal .
viii) The vibrations of guitar strings when played are
sinusoidal in nature.
24. Applications of Trigonometry
Graph of Trigonometric Functions
Graph of sine function
f(x) = a sin ( bx + c )
Graph of sine function
f(x) = a cos ( bx + c )
Graph of tangent function
f(x) = tanx
25. Applications of Trigonometry
The 17th and 18th centuries saw the invention of numerous
mechanical devices. A notable application was the science of artillery
—and in the 18th century it was a science. Galileo Galilei (1564–1642)
discovered that any motion—such as that of a projectile under the
force of gravity—can be resolved into two components, one horizontal
and the other vertical,
This discovery led scientists to the formula for the range of a
cannonball when its muzzle velocity v0 (the speed at which it leaves the
cannon) and the angle of elevation A of the cannon.
27. Applications of Trigonometry
Fourier series
An infinite trigonometric series of terms consisting of constants
multiplied by sines or cosines, used in the approximation of periodic
functions.
The trigonometric or Fourier series have found numerous applications
in almost every branch of science, from optics and acoustics to radio
transmission and earthquake analysis. Their extension to non
periodic functions played a key role in the development of quantum
mechanics in the early years of the 20th century. Trigonometry, by
and large, matured with Fourier's theorem.
28. Angle measurement
The concept of angle is one of the most important concepts in geometry
and the subject of trigonometry is based on the measurement of
angles.
Degree (Angle)
There are two commonly used units of measurement for angles. The
more familiar unit of measurement is that of degrees. A circle is
divided into 360 equal degrees,
Degrees may be further divided into minutes and seconds.
For instance seven and a half degrees is now usually written 7.5°.
Each degree is divided into 60 equal parts called minutes. So seven
and a half degrees can be called 7 degrees and 30 minutes, written 7°
30'. Each minute is further divided into 60 equal parts called seconds,
and, for instance, 2 degrees 5 minutes 30 seconds is written 2° 5' 30".
29. Angle measurement
Radian(Angle)
The other common measurement for angles is radians.
If the radius of the circle and the length of arc of a sector of the circle
are equal then angle is 1 radian.
The radian measure of the angle is the ratio of the length of the
subtended arc to the radius of the circle.
radian measure = arc length/radius (θ = S/r)
Below is a table of common angles in both
degree measurement and radian measurement.
Degrees Radians
90° Π /2
60° Π /3
45° Π /4
30° Π /6
31. Angle measurement
1. Express the following angles in radians.
(a). 12 degrees, 28 minutes, that is, 12° 28'. (b). 36° 12'.
2. Reduce the following numbers of radians to degrees, minutes, and
seconds.
(a). 0.47623. (b). 0.25412.
3. Given the angle a and the radius r, to find the length of the
subtending arc. a = 0° 17' 48", r = 6.2935.
4. Find the length to the nearest inch of a circular arc of 11 degrees
48.3 minutes if the radius is 3200 feet.
5. Given the length of the arc l and the angle a which it subtends at
the center, to find the radius.
a = 0° 44' 30", l = .032592
32. Properties of sines and cosines
1.Sine and cosine are periodic functions of period 360° or 2Π .,
sin (t + 360°) = sin t, and sin (t + 2π ) = sin t,
cos (t + 360°) = cos t. cos (t + 2π ) = cos t.
2 . Sine and cosine are complementary:
cos t = sin (π /2 – t) , sin t = cos (π /2 – t)
3 .The Pythagorean identity sin2
t + cos2
t = 1.
4. Sine is an odd function, and cosine is even
sin (–t) = –sin t, and cos (–t) = cos t.
5.An obvious property of sines and cosines is that their values lie
between –1 and 1. Every point on the unit circle is 1 unit from the
origin, so the coordinates of any point are within 1 of 0 as well.
33. The Law(Rule) of sines
The Law of Sines is simple and beautiful and easy to derive. It’s
useful when you know two angles and any side of a triangle, or
sometimes when you know two sides and one angle.
Law of Sines — First Form:
a / sin A = b / sin B = c / sin C
This is very simple and beautiful: for any triangle, if you divide
any of the three sides by the sine of the opposite angle, you’ll get
the same result. This law is valid for any triangle.
Law of Sines— Second Form:
sin A / a = sin B / b = sin C / c
34. The Law(Rules) of cosines
The Law of Sines is fine when you can relate sides and angles. But
suppose you know three sides of the triangle — for instance
a = 180, b = 238, c = 340 — and you have to find the three angles.
The Law of Sines is no good for that because it relates two sides
and their opposite angles. If you don’t know any angles, you have
an equation with two unknowns and you can’t solve it.
Law of Cosines — First Form:
cos A = (b² + c² − a²) / 2bc
cos B = (a² + c² − b²) / 2ac
cos C = (a² + b² − c²) / 2ab
Law of Cosines — Second Form:
a² = b² + c² − 2bc cos A
b² = a² + c² − 2ac cos B
c² = a² + b² − 2ab cos C
35. Trigonometric Equations
A formula that asserts that two expressions have the same value ;it is
either an identical equation or an identity which is true for any values of
the variables or a conditional equation which is only true for certain
values of the variables.
Example1
Solve the equation sin{1/3(θ-30)°} = √3/2, giving all the roots in the
interval 0°≤ θ ≤360°.
Example2
Find all the values of θ in the interval 0°≤ θ ≤360° for which
sin2θ = cos36°
Example3
Find all the values of θ in the interval 0°≤ θ ≤360° for which
sin2θ °-√ 3 cos2θ° = 0
36. Applications of Trigonometric Equations
Example
The height in meters of the water in a harbor is given by approximately
by the formula d=6+3cos30t° where t is the time measured in hours from
noon. Find the time after noon when the height of the water is 7.5 meters
for the second time.