The history of trigonometry is inextricably linked with astronomy, because it was to solve the problems of this science that ancient scientists began to study the relationships of various quantities in a triangle.
Today, trigonometry is a micro-branch of mathematics that studies the relationship between the values of the angles and lengths of the sides of triangles, and also deals with the analysis of algebraic identities of trigonometric functions.
The term "trigonometry"
The term itself, which gave its name to this branch of mathematics, was first discovered in the title of a book authored by the German mathematician Pitiscus in 1505. The word "trigonometry" is of Greek origin and means "measuring a triangle." To be more precise, we are not talking about the literal measurement of this figure, but about its solution, that is, determining the values of its unknown elements using known ones.
General information about trigonometry
The history of trigonometry began more than two thousand years ago. Initially, its emergence was associated with the need to clarify the relationships between the angles and sides of a triangle. In the process of research, it turned out that the mathematical expression of these relationships requires the introduction of special trigonometric functions, which were initially designed as numerical tables.
For many sciences related to mathematics, the impetus for development was the history of trigonometry. The origin of the units of measurement of angles (degrees), associated with the research of scientists of Ancient Babylon, is based on the sexagesimal notation system, which gave rise to the modern decimal notation used in many applied sciences.
It is assumed that trigonometry originally existed as a part of astronomy. Then it began to be used in architecture. And over time, the expediency of applying this science in various areas of human activity arose. These are, in particular, astronomy, sea and air navigation, acoustics, optics, electronics, architecture and others.
Trigonometry in the early centuries
Guided by data on surviving scientific relics, the researchers concluded that the history of trigonometry is connected with the work of the Greek astronomer Hipparchus, who first thought about finding ways to solve (spherical) triangles. His works date back to the 2nd century BC.
Also, one of the most important achievements of those times was the determination of the relationship between the legs and the hypotenuse in right triangles, which later became known as the Pythagorean theorem.
The history of the development of trigonometry in Ancient Greece is associated with the name of the astronomer Ptolemy - the author of the geocentric theory that dominated before Copernicus.
Greek astronomers did not know sines, cosines and tangents. They used tables that allowed them to find the value of the chord of a circle using a subtended arc. The units for measuring chords were degrees, minutes and seconds. One degree was equal to a sixtieth part of the radius.
Also, the research of the ancient Greeks advanced the development of spherical trigonometry. In particular, Euclid in his “Principles” gives a theorem about the patterns of relationships between the volumes of balls of different diameters. His works in this area became a kind of impetus for the development of related areas of knowledge. This is, in particular, the technology of astronomical instruments, the theory of map projections, the celestial coordinate system, etc.
Middle Ages: research by Indian scientists
Indian medieval astronomers achieved significant success. The death of ancient science in the 4th century led to the movement of the center of development of mathematics to India.
The history of the emergence of trigonometry as a separate section of mathematical teaching began in the Middle Ages. It was then that scientists replaced chords with sinuses. This discovery made it possible to introduce functions related to the study of sides and angles. That is, it was then that trigonometry began to separate itself from astronomy, turning into a branch of mathematics.
Aryabhata had the first tables of sines; they were drawn through 3 o, 4 o, 5 o. Later, detailed versions of the tables appeared: in particular, Bhaskara presented a table of sines in 1 o.
The first specialized treatise on trigonometry appeared in the 10th-11th centuries. Its author was the Central Asian scientist Al-Biruni. And in his main work, “The Canon of Mas‘ud” (Book III), the medieval author goes even deeper into trigonometry, giving a table of sines (in 15-inch increments) and a table of tangents (in 1° increments).
History of the development of trigonometry in Europe
After the translation of Arabic treatises into Latin (XII-XIII centuries), most of the ideas of Indian and Persian scientists were borrowed by European science. The first mentions of trigonometry in Europe date back to the 12th century.
According to researchers, the history of trigonometry in Europe is connected with the name of the Englishman Richard of Wallingford, who became the author of the essay “Four Treatises on Straight and Inverted Chords.” It was his work that became the first work entirely devoted to trigonometry. By the 15th century, many authors mentioned trigonometric functions in their works.
History of trigonometry: Modern times
In modern times, most scientists began to realize the extreme importance of trigonometry not only in astronomy and astrology, but also in other areas of life. These are, first of all, artillery, optics and navigation on long sea voyages. Therefore, in the second half of the 16th century, this topic interested many prominent people of that time, including Nicolaus Copernicus and Francois Vieta. Copernicus devoted several chapters to trigonometry in his treatise “On the Rotation of the Celestial Spheres” (1543). A little later, in the 60s of the 16th century, Rheticus, a student of Copernicus, cited fifteen-digit trigonometric tables in his work “The Optical Part of Astronomy”.
In the “Mathematical Canon” (1579) he gives a detailed and systematic, although unproven, characterization of plane and spherical trigonometry. And Albrecht Durer became the one thanks to whom the sine wave was born.
Merits of Leonhard Euler
Giving trigonometry modern content and form was the merit of Leonhard Euler. His treatise "An Introduction to the Analysis of Infinites" (1748) contains a definition of the term "trigonometric functions" that is equivalent to the modern one. Thus, this scientist was able to determine But that's not all.
The definition of trigonometric functions on the entire number line became possible thanks to Euler's research not only on permissible negative angles, but also on angles greater than 360°. It was he who first proved in his works that the cosine and tangent of a right angle are negative. The expansion of integer powers of cosine and sine was also the merit of this scientist. The general theory of trigonometric series and the study of the convergence of the resulting series were not the objects of Euler's research. However, while working on related problems, he made many discoveries in this area. It was thanks to his work that the history of trigonometry continued. In his works he briefly touched upon issues of spherical trigonometry.
Applications of trigonometry
Trigonometry is not an applied science; its problems are rarely used in real everyday life. However, this fact does not reduce its significance. Very important, for example, is the technique of triangulation, which allows astronomers to accurately measure the distance to nearby stars and monitor satellite navigation systems.
Trigonometry is also used in navigation, music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (for example, in decoding ultrasound examinations, ultrasound and computed tomography), pharmaceuticals, chemistry, number theory, seismology, meteorology , oceanology, cartography, many sections of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography, etc. The history of trigonometry and its role in the study of natural and mathematical sciences is still studied to this day. Perhaps in the future there will be even more areas of its application.
History of the origin of basic concepts
The history of the emergence and development of trigonometry goes back more than one century. The introduction of the concepts that form the basis of this section of mathematical science also did not happen overnight.
Thus, the concept of “sine” has a very long history. Mentions of various relationships between segments of triangles and circles are found in scientific works dating back to the 3rd century BC. The works of such great ancient scientists as Euclid, Archimedes, and Apollonius of Perga already contain the first studies of these relationships. New discoveries required certain terminological clarifications. Thus, the Indian scientist Aryabhata gives the chord the name “jiva”, meaning “bow string”. When Arabic mathematical texts were translated into Latin, the term was replaced by a similar meaning, sine (i.e., “bend”).
The word "cosine" appeared much later. The term is a shortened version of the Latin phrase "supplementary sine".
The emergence of tangents is associated with deciphering the problem of determining the length of the shadow. The term “tangent” was introduced in the 10th century by the Arab mathematician Abu-l-Wafa, who compiled the first tables for determining tangents and cotangents. But European scientists did not know about these achievements. The German mathematician and astronomer Regimontanus rediscovered these concepts in 1467. The proof of the tangent theorem is his merit. And this term is translated as “concerning.”
Trigonometry in astronomy:
The need for solving triangles was first discovered in astronomy; therefore, for a long time, trigonometry was developed and studied as one of the branches of astronomy.
The tables of the positions of the Sun and Moon compiled by Hipparchus made it possible to pre-calculate the moments of the onset of eclipses (with an error of 1-2 hours). Hipparchus was the first to use spherical trigonometry methods in astronomy. He increased the accuracy of his observations by using a cross of threads in goniometric instruments—sextants and quadrants—to point at the luminary. The scientist compiled a huge catalog of the positions of 850 stars for those times, dividing them by brightness into 6 degrees (stellar magnitudes). Hipparchus introduced geographical coordinates - latitude and longitude, and he can be considered the founder of mathematical geography. (c. 190 BC - c. 120 BC)
A complete solution to the problem of determining all the elements of a plane or spherical triangle from three given elements, important expansions of sinпх and cosпх in powers of cos x and sinx. Knowledge of the formula for sines and cosines of multiple arcs enabled Viet to solve the 45th degree equation proposed by the mathematician A. Roomen; Viète showed that the solution to this equation is reduced to dividing the angle into 45 equal parts and that there are 23 positive roots of this equation. Vieth solved Apollonius' problem using a ruler and compass.
Solving spherical triangles is one of the problems of astronomy. The following theorems allow us to calculate the sides and angles of any spherical triangle from three appropriately specified sides or angles: (sine theorem) (cosine theorem for angles) (cosine theorem for sides).
Trigonometry in physics:
types of oscillatory phenomena.
Harmonic oscillation is a phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity oscillates harmoniously and changes over time as follows:
Where x is the value of the changing quantity, t is time, A is the amplitude of oscillations, ω is the cyclic frequency of oscillations, is the full phase of oscillations, r is the initial phase of oscillations.
Mechanical vibrations . Mechanical vibrations
Trigonometry in nature.
We often ask the question
- One of fundamental properties
- - these are more or less regular changes in the nature and intensity of biological processes.
- Basic earth rhythm- daily allowance.
Trigonometry in biology
- Trigonometry plays an important role in medicine. With its help, Iranian scientists discovered the heart formula - a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 basic parameters, including several additional ones for calculations in cases of arrhythmia.
- diatonic scale 2:3:5
Trigonometry in architecture
- Swiss Re Insurance Corporation in London
- Interpretation
We have given only a small part of where you can find trigonometric functions. We found out
We have proven that trigonometry is closely related to physics and is found in nature and medicine. One can give endlessly many examples of periodic processes of living and inanimate nature. All periodic processes can be described using trigonometric functions and depicted on graphs
We think that trigonometry is reflected in our lives, and the spheres
in which it plays an important role will expand.
- Found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.
- Proved
- We think
View document contents
"Danilova T.V.-script"
MKOU "Nenets secondary school - boarding school named after. A.P. Pyrerki"
Educational project
" "
Danilova Tatyana Vladimirovna
Mathematic teacher
Justification of the relevance of the project.
Trigonometry is the branch of mathematics that studies trigonometric functions. It’s hard to imagine, but we encounter this science not only in mathematics lessons, but also in our everyday life. You might not have suspected it, but trigonometry is found in such sciences as physics, biology, it plays an important role in medicine, and, most interestingly, even music and architecture cannot do without it.
The word trigonometry first appears in 1505 in the title of a book by the German mathematician Pitiscus.
Trigonometry is a Greek word, and literally translated means the measurement of triangles (trigonan - triangle, metreo - I measure).
The emergence of trigonometry was closely related to land surveying, astronomy and construction.…
A schoolchild at the age of 14-15 does not always know where he will go to study and where he will work.
For some professions, its knowledge is necessary, because... allows you to measure distances to nearby stars in astronomy, between landmarks in geography, and control satellite navigation systems. The principles of trigonometry are also used in such areas as music theory, acoustics, optics, financial market analysis, electronics, probability theory, statistics, biology, medicine (including ultrasound and computed tomography), pharmaceuticals, chemistry, number theory ( and, as a consequence, cryptography), seismology, meteorology, oceanology, cartography, many branches of physics, topography and geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography.
Definition of the subject of research
3. Project goals.
Problematic question
1. Which trigonometry concepts are most often used in real life?
2. What role does trigonometry play in astronomy, physics, biology and medicine?
3. How are architecture, music and trigonometry related?
Hypothesis
Hypothesis testing
Trigonometry (from Greektrigonon - triangle,metro – metric) –
History of trigonometry:
Ancient people calculated the height of a tree by comparing the length of its shadow with the length of the shadow of a pole whose height was known. The stars were used to calculate the location of a ship at sea.
The next step in the development of trigonometry was made by the Indians in the period from the 5th to the 12th centuries.
The term cosine itself appeared much later in the works of European scientists for the first time at the end of the 16th century from the so-called “sine of the complement”, i.e. sine of the angle that complements the given angle to 90°. “Sine of the complement” or (in Latin) sinus complementi began to be abbreviated as sinus co or co-sinus.
In the XVII – XIX centuries. trigonometry becomes one of the chapters of mathematical analysis.
It finds wide application in mechanics, physics and technology, especially in the study of oscillatory movements and other periodic processes.
Jean Fourier proved that any periodic motion can be represented (with any degree of accuracy) as a sum of simple harmonic oscillations.
into the system of mathematical analysis.
Where is trigonometry used?
Trigonometric calculations are used in almost all areas of human life. It should be noted that it is used in such areas as astronomy, physics, nature, biology, music, medicine and many others.
Trigonometry in astronomy:
The need for solving triangles was first discovered in astronomy; therefore, for a long time, trigonometry was developed and studied as one of the branches of astronomy.
The need for solving triangles was first discovered in astronomy; therefore, for a long time, trigonometry was developed and studied as one of the branches of astronomy.
Vieta's achievements in trigonometry
A complete solution to the problem of determining all the elements of a plane or spherical triangle from three given elements, important expansions of sinпх and cosпх in powers of cos x and sinx. Knowledge of the formula for sines and cosines of multiple arcs enabled Viet to solve the 45th degree equation proposed by the mathematician A. Roomen; Viète showed that the solution to this equation is reduced to dividing the angle into 45 equal parts and that there are 23 positive roots of this equation. Vieth solved Apollonius' problem using a ruler and compass.
Solving spherical triangles is one of the problems of astronomy. The following theorems allow us to calculate the sides and angles of any spherical triangle from three appropriately specified sides or angles: (sine theorem) (cosine theorem for angles) (cosine theorem for sides).
Trigonometry in physics:
In the world around us we have to deal with periodic processes that repeat at regular intervals. These processes are called oscillatory. Oscillatory phenomena of various physical natures obey general laws and are described by the same equations. There are different types of oscillatory phenomena.
Harmonic oscillation- the phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity oscillates harmoniously and changes over time as follows:
Where x is the value of the changing quantity, t is time, A is the amplitude of oscillations, ω is the cyclic frequency of oscillations, is the full phase of oscillations, r is the initial phase of oscillations.
Generalized harmonic oscillation in differential form x’’ + ω²x = 0.
Mechanical vibrations . Mechanical vibrations are movements of bodies that repeat at exactly equal intervals of time. A graphical representation of this function gives a visual representation of the course of the oscillatory process over time. Examples of simple mechanical oscillatory systems are a weight on a spring or a mathematical pendulum.
Trigonometry in nature.
We often ask the question “Why do we sometimes see things that aren’t really there?”. The following questions are proposed for research: “How does a rainbow appear? Northern Lights?”, “What are optical illusions?” "How can trigonometry help answer these questions?"
The rainbow theory was first proposed in 1637 by Rene Descartes. He explained rainbows as a phenomenon related to the reflection and refraction of light in raindrops.
Northern Lights The penetration of charged solar wind particles into the upper layers of the atmosphere of planets is determined by the interaction of the planet’s magnetic field with the solar wind.
The force acting on a charged particle moving in a magnetic field is called the Lorentz force. It is proportional to the charge of the particle and the vector product of the field and the speed of the particle.
American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth and the plane of vision.
In addition, in biology such concepts as carotid sinus, carotid sinus and venous or cavernous sinus are used.
Trigonometry plays an important role in medicine. With its help, Iranian scientists discovered the heart formula - a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 basic parameters, including several additional ones for calculations in cases of arrhythmia.
One of fundamental properties living nature is the cyclical nature of most of the processes occurring in it.
Biological rhythms, biorhythms
Basic earth rhythm– daily allowance.
A model of biorhythms can be built using trigonometric functions.
Trigonometry in biology
What biological processes are associated with trigonometry?
Trigonometry plays an important role in medicine. With its help, Iranian scientists discovered the heart formula - a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 basic parameters, including several additional ones for calculations in cases of arrhythmia.
Biological rhythms, biorhythms are associated with trigonometry
A model of biorhythms can be built using graphs of trigonometric functions. To do this, you need to enter the person’s date of birth (day, month, year) and forecast duration
The movement of fish in water occurs according to the law of sine or cosine, if you fix a point on the tail and then consider the trajectory of movement.
The emergence of musical harmony
According to legends that have come down from ancient times, the first to try to do this were Pythagoras and his students.
Frequencies corresponding to the same note in the first, second, etc. octaves are related as 1:2:4:8...
diatonic scale 2:3:5
Trigonometry in architecture
Gaudi Children's School in Barcelona
Swiss Re Insurance Corporation in London
Felix Candela Restaurant in Los Manantiales
Interpretation
We have given only a small part of where trigonometric functions can be found. We found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.
We have proven that trigonometry is closely related to physics and is found in nature and medicine. One can give endlessly many examples of periodic processes of living and inanimate nature. All periodic processes can be described using trigonometric functions and depicted on graphs
We think that trigonometry is reflected in our lives, and the spheres
in which it plays an important role will expand.
Found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.
Proved that trigonometry is closely related to physics, found in nature, music, astronomy and medicine.
We think that trigonometry is reflected in our lives, and the areas in which it plays an important role will expand.
7. Literature.
Maple6 program that implements the image of graphs
"Wikipedia"
Ucheba.ru
Math.ru "library"
View presentation content
"Danilova T.V."
" Trigonometry in the world around us and human life "
Research objectives:
The connection between trigonometry and real life.
Problematic question 1. Which trigonometry concepts are most often used in real life? 2. What role does trigonometry play in astronomy, physics, biology and medicine? 3. How are architecture, music and trigonometry related?
Hypothesis
Most physical phenomena of nature, physiological processes, patterns in music and art can be described using trigonometry and trigonometric functions.
What is trigonometry???
Trigonometry (from the Greek trigonon - triangle, metro - metric) - microsection of mathematics, which studies the relationships between the values of angles and the lengths of the sides of triangles, as well as algebraic identities of trigonometric functions.
History of trigonometry
The origins of trigonometry date back to ancient Egypt, Babylonia and the Indus Valley over 3,000 years ago.
The word trigonometry first appears in 1505 in the title of a book by the German mathematician Pitiscus.
For the first time, methods for solving triangles based on the dependencies between the sides and angles of a triangle were found by the ancient Greek astronomers Hipparchus and Ptolemy.
Ancient people calculated the height of a tree by comparing the length of its shadow with the length of the shadow of a pole whose height was known.
The stars were used to calculate the location of a ship at sea.
The next step in the development of trigonometry was made by the Indians in the period from the 5th to the 12th centuries.
IN difference from the Greeks yians began to consider and use in calculations no longer the whole chord of MM the corresponding central angle, but only its half MR, i.e. sine - half of the central angle.
The term cosine itself appeared much later in the works of European scientists for the first time at the end of the 16th century from the so-called « sine's complement » , i.e. sine of the angle that complements the given angle to 90 . « Sine complement » or (in Latin) sinus complementi began to be abbreviated as sinus co or co-sinus.
Along with the sine, the Indians introduced into trigonometry cosine , more precisely, they began to use the cosine line in their calculations. They also knew the relations cos =sin(90 - ) and sin 2 +cos 2 =r 2 , as well as formulas for the sine of the sum and difference of two angles.
In the XVII – XIX centuries. trigonometry becomes
one of the chapters of mathematical analysis.
It finds wide application in mechanics,
physics and technology, especially when studying
oscillatory movements and others
periodic processes.
Viète, whose first mathematical studies related to trigonometry, knew about the properties of periodicity of trigonometric functions.
Proved that every periodic
movement may be
presented (with any degree
accuracy) in the form of a sum of primes
harmonic vibrations.
Founder analytical
theories
trigonometric functions .
Leonard Euler
In "Introduction to the Analysis of Infinites" (1748)
interprets sine, cosine, etc. not like
trigonometric lines, required
related to the circle, and how
trigonometric functions that he
viewed as a relationship between the parties
right triangle like numbers
quantities.
Excluded from my formulas
R – whole sine, taking
R = 1, and simplified it like this
way of recording and calculation.
Develops doctrine
about trigonometric functions
any argument.
Continued in the 19th century
theory development
trigonometric
functions.
N.I.Lobachevsky
“Geometric considerations,” writes Lobachevsky, “are necessary until the beginning of trigonometry, until they serve to discover the distinctive properties of trigonometric functions... From here, trigonometry becomes completely independent of geometry and has all the advantages of analysis.”
Stages of development of trigonometry:
- Trigonometry was brought to life by the need to measure angles.
- The first steps of trigonometry were to establish connections between the magnitude of the angle and the ratio of specially constructed straight line segments. The result is the ability to solve planar triangles.
- The need to tabulate the values of entered trigonometric functions.
- Trigonometric functions turned into independent objects of research.
- In the 18th century trigonometric functions were included
into the system of mathematical analysis.
Where is trigonometry used?
Trigonometric calculations are used in almost all areas of human life. It should be noted that it is used in such areas as astronomy, physics, nature, biology, music, medicine and many others.
Trigonometry in astronomy
The need for solving triangles was first discovered in astronomy; therefore, for a long time, trigonometry was developed and studied as one of the branches of astronomy.
Trigonometry also reached significant heights among Indian medieval astronomers.
The main achievement of Indian astronomers was the replacement of chords
sines, which made it possible to introduce various functions related
with the sides and angles of a right triangle.
Thus, the beginning of trigonometry was laid in India
as the study of trigonometric quantities.
The tables of the positions of the Sun and Moon compiled by Hipparchus made it possible to pre-calculate the moments of the onset of eclipses (with an error of 1-2 hours). Hipparchus was the first to use spherical trigonometry methods in astronomy. He increased the accuracy of observations by using a cross of threads in goniometric instruments - sextants and quadrants - to point at the luminary. The scientist compiled a huge catalog of the positions of 850 stars for those times, dividing them by brightness into 6 degrees (stellar magnitudes). Hipparchus introduced geographical coordinates - latitude and longitude, and he can be considered the founder of mathematical geography. (c. 190 BC - c. 120 BC)
Hipparchus
Trigonometry in physics
In the world around us we have to deal with periodic processes that repeat at regular intervals. These processes are called oscillatory. Oscillatory phenomena of various physical natures obey general laws and are described by the same equations. There are different types of oscillatory phenomena, for example:
Mechanical vibrations
Harmonic vibrations
Harmonic vibrations
Harmonic oscillation - the phenomenon of periodic change of any quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity oscillates harmoniously and changes over time as follows:
or
Where x is the value of the changing quantity, t is time, A is the amplitude of oscillations, ω is the cyclic frequency of oscillations, is the full phase of oscillations, r is the initial phase of oscillations.
Generalized harmonic oscillation in differential form x’’ + ω²x = 0.
Mechanical vibrations
Mechanical vibrations are movements of bodies that repeat at exactly equal intervals of time. A graphical representation of this function gives a visual representation of the course of the oscillatory process over time.
Examples of simple mechanical oscillatory systems are a weight on a spring or a mathematical pendulum.
Math pendulum
The figure shows the oscillations of a pendulum; it moves along a curve called cosine.
Bullet trajectory and vector projections on the X and Y axes
The figure shows that the projections of the vectors on the X and Y axes are respectively equal
υ x = υ o cos α
υ y = υ o sin α
Trigonometry in nature
We often ask the question “Why do we sometimes see things that aren’t really there?”. The following questions are proposed for research: “How does a rainbow appear? Northern Lights?”, “What are optical illusions?” "How can trigonometry help answer these questions?"
Optical illusions
natural
artificial
mixed
Rainbow theory
Rainbows occur when sunlight is refracted by water droplets suspended in the air. law of refraction:
The rainbow theory was first proposed in 1637 by Rene Descartes. He explained rainbows as a phenomenon related to the reflection and refraction of light in raindrops.
sin α /sin β = n 1 /n 2
where n 1 =1, n 2 ≈1.33 are the refractive indices of air and water, respectively, α is the angle of incidence, and β is the angle of refraction of light.
Northern lights
The penetration of charged solar wind particles into the upper atmosphere of planets is determined by the interaction of the planet’s magnetic field with the solar wind.
The force acting on a charged particle moving in a magnetic field is called the Lorentz force. It is proportional to the charge of the particle and the vector product of the field and the speed of the particle.
- American scientists claim that the brain estimates the distance to objects by measuring the angle between the plane of the earth and the plane of vision.
- In addition, in biology such concepts as carotid sinus, carotid sinus and venous or cavernous sinus are used.
- Trigonometry plays an important role in medicine. With its help, Iranian scientists discovered the heart formula - a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 basic parameters, including several additional ones for calculations in cases of arrhythmia.
- One of fundamental properties living nature is the cyclical nature of most of the processes occurring in it.
- Biological rhythms, biorhythms– these are more or less regular changes in the nature and intensity of biological processes.
- Basic earth rhythm– daily allowance.
- A model of biorhythms can be built using trigonometric functions.
Trigonometry in biology
What biological processes are associated with trigonometry?
- Trigonometry plays an important role in medicine. With its help, Iranian scientists discovered the heart formula - a complex algebraic-trigonometric equation consisting of 8 expressions, 32 coefficients and 33 basic parameters, including several additional ones for calculations in cases of arrhythmia.
- Biological rhythms, biorhythms are associated with trigonometry.
- A model of biorhythms can be built using graphs of trigonometric functions.
- To do this, you need to enter the person’s date of birth (day, month, year) and the duration of the forecast.
Trigonometry in biology
The movement of fish in water occurs according to the law of sine or cosine, if you fix a point on the tail and then consider the trajectory of movement.
When swimming, the fish's body takes the shape of a curve that resembles the graph of the function y=tgx.
The emergence of musical harmony
- According to legends that have come down from ancient times, the first to try to do this were Pythagoras and his students.
- Frequencies corresponding
the same note in the first, second, etc. octaves are related as 1:2:4:8...
- diatonic scale 2:3:5
Music has its own geometry
Tetrahedron of different types of chords of four sounds:
blue – small intervals;
warmer tones - more “discharged” chord sounds; The red sphere is the most harmonious chord with equal intervals between notes.
cos 2 C + sin 2 C = 1
AC– the distance from the top of the statue to the person’s eyes,
AN– height of the statue,
sin C- sine of the angle of incidence of gaze.
Trigonometry in architecture
Gaudi Children's School in Barcelona
Swiss Re Insurance Corporation in London
y = f (λ)cos θ
z = f (λ)sin θ
Felix Candela Restaurant in Los Manantiales
- Found out that trigonometry was brought to life by the need to measure angles, but over time it developed into the science of trigonometric functions.
- Proved that trigonometry is closely related to physics, found in nature, music, astronomy and medicine.
- We think that trigonometry is reflected in our lives, and the areas in which it plays an important role will expand.
Trigonometry has come a long way in development. And now, we can say with confidence that trigonometry does not depend on other sciences, and other sciences depend on trigonometry.
- Maslova T.N. "Student's Guide to Mathematics"
- Maple6 program that implements the image of graphs
- "Wikipedia"
- Ucheba.ru
- Math.ru "library"
- History of mathematics from ancient times to the beginning of the 19th century in 3 volumes // ed. A. P. Yushkevich. Moscow, 1970 – volume 1-3 E. T. Bell Creators of mathematics.
- Predecessors of modern mathematics // ed. S. N. Niro. Moscow, 1983 A. N. Tikhonov, D. P. Kostomarov.
- Stories about applied mathematics//Moscow, 1979. A.V. Voloshinov. Mathematics and art // Moscow, 1992. Newspaper Mathematics. Supplement to the newspaper dated September 1, 1998.
1. Trigonometric functions are elementary functions whose argument is corner. Trigonometric functions describe the relationships between sides and acute angles in a right triangle. The areas of application of trigonometric functions are extremely diverse. For example, any periodic processes can be represented as a sum of trigonometric functions (Fourier series). These functions often appear when solving differential and functional equations.
2. Trigonometric functions include the following 6 functions: sinus, cosine, tangent,cotangent, secant And cosecant. For each of these functions there is an inverse trigonometric function.
3. It is convenient to introduce the geometric definition of trigonometric functions using unit circle. The figure below shows a circle with radius r=1. The point M(x,y) is marked on the circle. The angle between the radius vector OM and the positive direction of the Ox axis is equal to α.
4. Sinus angle α is the ratio of the ordinate y of the point M(x,y) to the radius r:
sinα=y/r.
Since r=1, then the sine is equal to the ordinate of the point M(x,y).
5. Cosine angle α is the ratio of the abscissa x of the point M(x,y) to the radius r:
cosα=x/r
6. Tangent angle α is the ratio of the ordinate y of a point M(x,y) to its abscissa x:
tanα=y/x,x≠0
7. Cotangent angle α is the ratio of the abscissa x of a point M(x,y) to its ordinate y:
cotα=x/y,y≠0
8. Secant angle α is the ratio of the radius r to the abscissa x of the point M(x,y):
secα=r/x=1/x,x≠0
9. Cosecant angle α is the ratio of the radius r to the ordinate y of the point M(x,y):
cscα=r/y=1/y,y≠0
10. In the unit circle, the projections x, y, the points M(x,y) and the radius r form a right triangle, in which x,y are the legs, and r is the hypotenuse. Therefore, the above definitions of trigonometric functions as applied to a right triangle are formulated as follows:
Sinus angle α is the ratio of the opposite side to the hypotenuse.
Cosine angle α is the ratio of the adjacent leg to the hypotenuse.
Tangent angle α is called the opposite leg to the adjacent one.
Cotangent angle α is called the adjacent side to the opposite side.
Secant angle α is the ratio of the hypotenuse to the adjacent leg.
Cosecant angle α is the ratio of the hypotenuse to the opposite leg.
11. Graph of the sine function
y=sinx, domain of definition: x∈R, range of values: −1≤sinx≤1
12. Graph of the cosine function
y=cosx, domain: x∈R, range: −1≤cosx≤1
13. Graph of the tangent function 14. Graph of the cotangent function 15. Graph of the secant function
y=tanx, range of definition: x∈R,x≠(2k+1)π/2, range of values: −∞ 
y=cotx, domain: x∈R,x≠kπ, range: −∞ 
y=secx, domain: x∈R,x≠(2k+1)π/2, range: secx∈(−∞,−1]∪∪)