Trigonometry is a branch of mathematical science that studies trigonometric functions and their use in geometry. The development of trigonometry began in ancient Greece. During the Middle Ages important contribution Scientists from the Middle East and India contributed to the development of this science.
This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the main trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.
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Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of aspect ratios right triangle.
Definitions of trigonometric functions
The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.
Cosine of the angle (cos α) - the ratio of the adjacent leg to the hypotenuse.
Angle tangent (t g α) - ratio opposite leg to the adjacent one.
Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.
These definitions are given for the acute angle of a right triangle!
Let's give an illustration.
IN triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.
The definitions of sine, cosine, tangent and cotangent allow you to calculate the values of these functions from the known lengths of the sides of the triangle.
Important to remember!
The range of values of sine and cosine is from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of values of tangent and cotangent is the entire number line, that is, these functions can take on any values.
The definitions given above apply to acute angles. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to 0 to 90 degrees. The rotation angle in degrees or radians is expressed by any real number from - ∞ to + ∞.
In this context, we can define sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Let's imagine unit circle centered at the origin of the Cartesian coordinate system.
The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).
Sine (sin) of the rotation angle
The sine of the rotation angle α is the ordinate of point A 1 (x, y). sin α = y
Cosine (cos) of the rotation angle
The cosine of the rotation angle α is the abscissa of point A 1 (x, y). cos α = x
Tangent (tg) of the rotation angle
The tangent of the angle of rotation α is the ratio of the ordinate of point A 1 (x, y) to its abscissa. t g α = y x
Cotangent (ctg) of the rotation angle
The cotangent of the rotation angle α is the ratio of the abscissa of point A 1 (x, y) to its ordinate. c t g α = x y
Sine and cosine are defined for any rotation angle. This is logical, because the abscissa and ordinate of a point after rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is undefined when a point after rotation goes to a point with zero abscissa (0, 1) and (0, - 1). In such cases, the expression for tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with cotangent. The difference is that the cotangent is not defined in cases where the ordinate of a point goes to zero.
Important to remember!
Sine and cosine are defined for any angles α.
Tangent is defined for all angles except α = 90° + 180° k, k ∈ Z (α = π 2 + π k, k ∈ Z)
Cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)
When deciding practical examples do not say "sine of the angle of rotation α". The words “angle of rotation” are simply omitted, implying that it is already clear from the context what is being discussed.
Numbers
What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?
Sine, cosine, tangent, cotangent of a number
Sine, cosine, tangent and cotangent of a number t is a number that is respectively equal to sine, cosine, tangent and cotangent in t radian.
For example, the sine of the number 10 π equal to sine rotation angle of 10 π rad.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's take a closer look at it.
Anyone real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.
The starting point on the circle is point A with coordinates (1, 0).
Positive number t
Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and will go the way t.
Now that the connection between a number and a point on a circle has been established, we move on to the definition of sine, cosine, tangent and cotangent.
Sine (sin) of t
Sine of a number t- ordinate of a point on the unit circle corresponding to the number t. sin t = y
Cosine (cos) of t
Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x
Tangent (tg) of t
Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t
The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. Point on the circle corresponding to the number t, coincides with the point to which the starting point goes after turning by an angle t radian.
Trigonometric functions of angular and numeric argument
Each value of angle α corresponds to specific value the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except α = 180° k, k ∈ Z (α = π k, k ∈ Z).
We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.
Similarly, we can talk about sine, cosine, tangent and cotangent as functions numeric argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.
Basic functions of trigonometry
Sine, cosine, tangent and cotangent are the basic trigonometric functions.
It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.
Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. Trigonometric definitions sine, cosine, tangent and cotangent are completely consistent with geometric definitions, given using the aspect ratios of a right triangle. Let's show it.
Take a unit circle with center at a rectangular Cartesian system coordinates Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right triangle, angle A 1 O H equal to angle turn α, the length of the leg O H is equal to the abscissa of point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.
In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side to the hypotenuse.
sin α = A 1 H O A 1 = y 1 = y
This means that determining the sine of an acute angle in a right triangle through the aspect ratio is equivalent to determining the sine of the rotation angle α, with alpha lying in the range from 0 to 90 degrees.
Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.
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The concepts of sine, cosine, tangent and cotangent are the main categories of trigonometry, a branch of mathematics, and are inextricably linked with the definition of angle. Ownership of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. That is why schoolchildren and students trigonometric calculations often cause difficulties. To overcome them, you should become more familiar with trigonometric functions and formulas.
Concepts in trigonometry
To understand basic concepts trigonometry, you must first decide what a right triangle and an angle in a circle are, and why all the basic trigonometric calculations are associated with them. A triangle in which one of the angles measures 90 degrees is rectangular. Historically, this figure was often used by people in architecture, navigation, art, and astronomy. Accordingly, by studying and analyzing the properties of this figure, people came to calculate the corresponding ratios of its parameters.
The main categories associated with right triangles are the hypotenuse and the legs. Hypotenuse - the side of a triangle opposite right angle. The legs, respectively, are the remaining two sides. The sum of the angles of any triangles is always 180 degrees.
Spherical trigonometry is a section of trigonometry that is not studied in school, but in applied sciences such as astronomy and geodesy, scientists use it. Feature of a triangle in spherical trigonometry is that it always has a sum of angles greater than 180 degrees.
Angles of a triangle
In a right triangle, the sine of an angle is the ratio of the leg opposite the desired angle to the hypotenuse of the triangle. Accordingly, cosine is the ratio of the adjacent leg and the hypotenuse. Both of these values always have a magnitude less than one, since the hypotenuse is always longer than the leg.
Tangent of an angle - value, equal to the ratio opposite side to adjacent leg the desired angle, or sine to cosine. Cotangent, in turn, is the ratio of the adjacent side of the desired angle to the opposite side. The cotangent of an angle can also be obtained by dividing one by the tangent value.
Unit circle
A unit circle in geometry is a circle whose radius equal to one. Such a circle is constructed in a Cartesian coordinate system, with the center of the circle coinciding with the origin point, and starting position The radius vector is determined by the positive direction of the X axis (abscissa axis). Each point on the circle has two coordinates: XX and YY, that is, the coordinates of the abscissa and ordinate. By selecting any point on the circle in the XX plane and dropping a perpendicular from it to the abscissa axis, we obtain a right triangle formed by the radius to the selected point (denoted by the letter C), the perpendicular drawn to the X axis (the intersection point is denoted by the letter G), and the segment the abscissa axis between the origin (the point is designated by the letter A) and the intersection point G. The resulting triangle ACG is a right triangle inscribed in a circle, where AG is the hypotenuse, and AC and GC are the legs. The angle between the radius of the circle AC and the segment of the abscissa axis with the designation AG is defined as α (alpha). So, cos α = AG/AC. Considering that AC is the radius of the unit circle, and it is equal to one, it turns out that cos α=AG. Likewise, sin α=CG.
In addition, knowing this data, you can determine the coordinate of point C on the circle, since cos α=AG, and sin α=CG, which means point C has given coordinates(cos α;sin α). Knowing that the tangent is equal to the ratio of sine to cosine, we can determine that tan α = y/x, and cot α = x/y. By considering angles in a negative coordinate system, you can calculate that the sine and cosine values of some angles can be negative.
Calculations and basic formulas
Trigonometric function values
Having considered the essence of trigonometric functions through the unit circle, we can derive the values of these functions for some angles. The values are listed in the table below.
The simplest trigonometric identities
Equations in which the sign of the trigonometric function contains unknown value, are called trigonometric. Identities with sin value x = α, k — any integer:
- sin x = 0, x = πk.
- 2. sin x = 1, x = π/2 + 2πk.
- sin x = -1, x = -π/2 + 2πk.
- sin x = a, |a| > 1, no solutions.
- sin x = a, |a| ≦ 1, x = (-1)^k * arcsin α + πk.
Identities with the value cos x = a, where k is any integer:
- cos x = 0, x = π/2 + πk.
- cos x = 1, x = 2πk.
- cos x = -1, x = π + 2πk.
- cos x = a, |a| > 1, no solutions.
- cos x = a, |a| ≦ 1, x = ±arccos α + 2πk.
Identities with the value tg x = a, where k is any integer:
- tan x = 0, x = π/2 + πk.
- tan x = a, x = arctan α + πk.
Identities with the value ctg x = a, where k is any integer:
- cot x = 0, x = π/2 + πk.
- ctg x = a, x = arcctg α + πk.
Reduction formulas
This category constant formulas denotes methods by which you can move from trigonometric functions of the form to functions of an argument, that is, reduce the sine, cosine, tangent and cotangent of an angle of any value to the corresponding indicators of the angle of the interval from 0 to 90 degrees for greater convenience of calculations.
Formulas for reducing functions for the sine of an angle look like this:
- sin(900 - α) = α;
- sin(900 + α) = cos α;
- sin(1800 - α) = sin α;
- sin(1800 + α) = -sin α;
- sin(2700 - α) = -cos α;
- sin(2700 + α) = -cos α;
- sin(3600 - α) = -sin α;
- sin(3600 + α) = sin α.
For cosine of angle:
- cos(900 - α) = sin α;
- cos(900 + α) = -sin α;
- cos(1800 - α) = -cos α;
- cos(1800 + α) = -cos α;
- cos(2700 - α) = -sin α;
- cos(2700 + α) = sin α;
- cos(3600 - α) = cos α;
- cos(3600 + α) = cos α.
The use of the above formulas is possible subject to two rules. First, if the angle can be represented as a value (π/2 ± a) or (3π/2 ± a), the value of the function changes:
- from sin to cos;
- from cos to sin;
- from tg to ctg;
- from ctg to tg.
The value of the function remains unchanged if the angle can be represented as (π ± a) or (2π ± a).
Secondly, the sign of the reduced function does not change: if it was initially positive, it remains so. Same with negative functions.
Addition formulas
These formulas express the values of sine, cosine, tangent and cotangent of the sum and difference of two rotation angles through their trigonometric functions. Typically the angles are denoted as α and β.
The formulas look like this:
- sin(α ± β) = sin α * cos β ± cos α * sin.
- cos(α ± β) = cos α * cos β ∓ sin α * sin.
- tan(α ± β) = (tg α ± tan β) / (1 ∓ tan α * tan β).
- ctg(α ± β) = (-1 ± ctg α * ctg β) / (ctg α ± ctg β).
These formulas are valid for any angles α and β.
Double and triple angle formulas
The double and triple angle trigonometric formulas are formulas that relate the functions of the angles 2α and 3α, respectively, to the trigonometric functions of the angle α. Derived from addition formulas:
- sin2α = 2sinα*cosα.
- cos2α = 1 - 2sin^2 α.
- tan2α = 2tgα / (1 - tan^2 α).
- sin3α = 3sinα - 4sin^3 α.
- cos3α = 4cos^3 α - 3cosα.
- tg3α = (3tgα - tg^3 α) / (1-tg^2 α).
Transition from sum to product
Considering that 2sinx*cosy = sin(x+y) + sin(x-y), simplifying this formula, we get identity sinα + sinβ = 2sin(α + β)/2 * cos(α − β)/2. Similarly sinα - sinβ = 2sin(α - β)/2 * cos(α + β)/2; cosα + cosβ = 2cos(α + β)/2 * cos(α − β)/2; cosα — cosβ = 2sin(α + β)/2 * sin(α − β)/2; tanα + tanβ = sin(α + β) / cosα * cosβ; tgα - tgβ = sin(α - β) / cosα * cosβ; cosα + sinα = √2sin(π/4 ∓ α) = √2cos(π/4 ± α).
Transition from product to sum
These formulas follow from the identities of the transition of a sum to a product:
- sinα * sinβ = 1/2*;
- cosα * cosβ = 1/2*;
- sinα * cosβ = 1/2*.
Degree reduction formulas
In these identities, the square and cubic powers of sine and cosine can be expressed in terms of the sine and cosine of the first power of a multiple angle:
- sin^2 α = (1 - cos2α)/2;
- cos^2 α = (1 + cos2α)/2;
- sin^3 α = (3 * sinα - sin3α)/4;
- cos^3 α = (3 * cosα + cos3α)/4;
- sin^4 α = (3 - 4cos2α + cos4α)/8;
- cos^4 α = (3 + 4cos2α + cos4α)/8.
Universal substitution
Formulas for universal trigonometric substitution express trigonometric functions in terms of the tangent of a half angle.
- sin x = (2tgx/2) * (1 + tan^2 x/2), with x = π + 2πn;
- cos x = (1 - tan^2 x/2) / (1 + tan^2 x/2), where x = π + 2πn;
- tg x = (2tgx/2) / (1 - tg^2 x/2), where x = π + 2πn;
- cot x = (1 - tg^2 x/2) / (2tgx/2), with x = π + 2πn.
Special cases
Special cases of protozoa trigonometric equations are given below (k is any integer).
Quotients for sine:
| Sin x value | x value |
|---|---|
| 0 | πk |
| 1 | π/2 + 2πk |
| -1 | -π/2 + 2πk |
| 1/2 | π/6 + 2πk or 5π/6 + 2πk |
| -1/2 | -π/6 + 2πk or -5π/6 + 2πk |
| √2/2 | π/4 + 2πk or 3π/4 + 2πk |
| -√2/2 | -π/4 + 2πk or -3π/4 + 2πk |
| √3/2 | π/3 + 2πk or 2π/3 + 2πk |
| -√3/2 | -π/3 + 2πk or -2π/3 + 2πk |
Quotients for cosine:
| cos x value | x value |
|---|---|
| 0 | π/2 + 2πk |
| 1 | 2πk |
| -1 | 2 + 2πk |
| 1/2 | ±π/3 + 2πk |
| -1/2 | ±2π/3 + 2πk |
| √2/2 | ±π/4 + 2πk |
| -√2/2 | ±3π/4 + 2πk |
| √3/2 | ±π/6 + 2πk |
| -√3/2 | ±5π/6 + 2πk |
Quotients for tangent:
| tg x value | x value |
|---|---|
| 0 | πk |
| 1 | π/4 + πk |
| -1 | -π/4 + πk |
| √3/3 | π/6 + πk |
| -√3/3 | -π/6 + πk |
| √3 | π/3 + πk |
| -√3 | -π/3 + πk |
Quotients for cotangent:
| ctg x value | x value |
|---|---|
| 0 | π/2 + πk |
| 1 | π/4 + πk |
| -1 | -π/4 + πk |
| √3 | π/6 + πk |
| -√3 | -π/3 + πk |
| √3/3 | π/3 + πk |
| -√3/3 | -π/3 + πk |
Theorems
Theorem of sines
There are two versions of the theorem - simple and extended. Simple theorem sines: a/sin α = b/sin β = c/sin γ. In this case, a, b, c are the sides of the triangle, and α, β, γ are the opposite angles, respectively.
Extended sine theorem for arbitrary triangle: a/sin α = b/sin β = c/sin γ = 2R. In this identity, R denotes the radius of the circle in which the given triangle is inscribed.
Cosine theorem
The identity is displayed as follows: a^2 = b^2 + c^2 - 2*b*c*cos α. In the formula, a, b, c are the sides of the triangle, and α is the angle opposite to side a.
Tangent theorem
The formula expresses the relationship between the tangents of two angles and the length of the sides opposite them. The sides are labeled a, b, c, and the corresponding opposite angles are α, β, γ. Formula of the tangent theorem: (a - b) / (a+b) = tan((α - β)/2) / tan((α + β)/2).
Cotangent theorem
Connects the radius of a circle inscribed in a triangle with the length of its sides. If a, b, c are the sides of the triangle, and A, B, C, respectively, are the angles opposite them, r is the radius of the inscribed circle, and p is the semi-perimeter of the triangle, the following identities are valid:
- cot A/2 = (p-a)/r;
- cot B/2 = (p-b)/r;
- cot C/2 = (p-c)/r.
Application
Trigonometry - not only theoretical science related to mathematical formulas. Its properties, theorems and rules are used in practice by various industries. human activity- astronomy, aerial and sea navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measuring work, computer graphics, cartography, oceanography, and many others.
Sine, cosine, tangent and cotangent are the basic concepts of trigonometry, with the help of which one can mathematically express the relationships between the angles and lengths of the sides in a triangle, and find the required quantities through identities, theorems and rules.
Lesson on the topic “Sine, cosine and tangent of an acute angle of a right triangle”
Lesson objectives:
educational - introduce the concept of sine, cosine, tangent of an acute angle in a right triangle, explore the dependencies and relationships between these quantities;
developing - the formation of the concept of sine, cosine, tangent as functions of an angle, the domain of definition of trigonometric functions, development logical thinking, development of correct mathematical speech;
educational – development of skills of independent work, culture of behavior, accuracy in record keeping.
Lesson progress:
“Education is not the number of lessons taken, but the number of understood. So, if you want to go forward, then hurry up slowly and be careful."
2. Lesson motivation.
One wise man said: “ Supreme manifestation spirit is the mind. The highest manifestation of reason is geometry. The geometry cell is a triangle. It is as inexhaustible as the Universe. The circle is the soul of geometry. Know the circle, and you will not only know the soul of geometry, but you will elevate your soul.”
We will try to do a little research together with you. Let's share your ideas that come to your mind, and don't be afraid to make mistakes, any thought can give us a new direction to search. Our achievements may not seem great to someone, but they will be our own achievements!
3. Updating of basic knowledge.
What angles can there be?
What are triangles?
What are the main elements that define a triangle?
What types of triangles are there depending on the sides?
What types of triangles are there depending on the angles?
What is a leg?
What is a hypotenuse?
What are the sides of a right triangle called?
What relationships between the sides and angles of this triangle do you know?
Why do you need to know the relationships between sides and angles?
What tasks in life may lead to the need to calculate unknown parties in a triangle?
The term "hypotenuse" comes from Greek word“hypoinouse”, meaning “stretching over something”, “contracting”. The word originates from the image of ancient Greek harps, on which the strings are stretched at the ends of two mutually perpendicular stands. The term "cathetus" comes from the Greek word "kathetos", which means the beginning of a "plumb line", "perpendicular".
Euclid said: “The legs are the sides that enclose a right angle.”
IN Ancient Greece a method for constructing a right triangle on the ground was already known. To do this, they used a rope on which 13 knots were tied, at the same distance from each other. During the construction of the pyramids in Egypt, right triangles were made in this way. This is probably why a right triangle with sides 3,4,5 was called Egyptian triangle.
4. Studying new material.
In ancient times, people watched the stars and, based on these observations, kept a calendar, calculated sowing dates, and the time of river floods; ships at sea and caravans on land navigated their journey by the stars. All this led to the need to learn how to calculate the sides in a triangle, two of whose vertices are on the ground, and the third is represented by a point in the starry sky. Based on this need, the science of trigonometry arose - a science that studies the connections between the sides of a triangle.
Do you think the relationships we already know are enough to solve such problems?
The purpose of today's lesson is to explore new connections and dependencies, to derive relationships, using which in the next geometry lessons you will be able to solve such problems.
Let's feel like we're in the role scientific workers and following the geniuses of antiquity Thales, Euclid, Pythagoras let's walk the path search for truth.
For this we need theoretical basis.
Highlight angle A and leg BC in red.
Highlight green leg AC.
Let's calculate what part is the opposite side for an acute angle A to its hypotenuse; to do this, we compose the ratio of the opposite side to the hypotenuse:
This relationship has a special name - such that every person in every point of the planet understands that we're talking about about a number representing the ratio of the opposite side of an acute angle to the hypotenuse. This word is sine. Write it down. Since the word sine without the name of the angle loses all meaning, the mathematical notation is as follows:
Now compose the ratio of the adjacent leg to the hypotenuse for acute angle A:
This ratio is called cosine. Its mathematical notation:
Let's consider another ratio for an acute angle A: the ratio of the opposite side to the adjacent side:
This ratio is called tangent. Its mathematical notation:
5. Consolidation of new material.
Let's consolidate our intermediate discoveries.
Sine is...
Cosine is...
Tangent is...
| sin A = | sin ABOUT = | sin A 1 = |
| cos A = | cos ABOUT = | cos A 1 = |
| tan A = | tg ABOUT = | tan A 1 = |
Solve orally No. 88, 889, 892 (work in pairs).
Using the acquired knowledge to solve practical problem:
“From the lighthouse tower, 70 m high, a ship is visible at an angle of 3° to the horizon. What's it like
distance from the lighthouse to the ship?
The problem is solved frontally. During the discussion, we make a drawing and the necessary notes on the board and in notebooks.
When solving the problem, Bradis tables are used.
Consider the solution to the problem p.175.
Solve No. 902(1).
6. Exercise for the eyes.
Without turning your head, look around the classroom wall around the perimeter clockwise, the chalkboard around the perimeter counterclockwise, the triangle depicted on the stand clockwise and the equal triangle counterclockwise. Turn your head to the left and look at the horizon line, and now at the tip of your nose. Close your eyes, count to 5, open your eyes and...
We'll put our palms to our eyes,
Let's spread our strong legs.
Turning to the right
Let's look around majestically.
And you need to go left too
Look from under your palms.
And - to the right! And further
Over your left shoulder!
Now let's continue working.
7. Independent work students.
Solve no.
8. Lesson summary. Reflection. D/z.
What new things have you learned? At the lesson:
have you considered...
you analyzed...
You received …
you have concluded...
you have replenished lexicon the following terms...
World science began with geometry. A person cannot truly develop culturally and spiritually if he has not studied geometry at school. Geometry arose not only from the practical, but also from the spiritual needs of man.
This is how she poetically explained her love for geometry
I love geometry...
I teach geometry because I love it
We need geometry, without it we can’t get anywhere.
Sine, cosine, circumference - everything is important here,
Everything is needed here
You just need to learn and understand everything very clearly,
Complete assignments and tests on time.
Trigonometry, as a science, originated in the Ancient East. First trigonometric ratios were deduced by astronomers to create accurate calendar and navigation by the stars. These calculations related to spherical trigonometry, while in school course study the ratios of sides and angles of a plane triangle.
Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationships between the sides and angles of triangles.
During the heyday of culture and science in the 1st millennium AD, knowledge spread from Ancient East to Greece. But the main discoveries of trigonometry are the merit of husbands Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.
Basic quantities of trigonometry
The basic trigonometric functions of a numeric argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.
The formulas for calculating the values of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants are equal in all directions,” since the proof is given using the example of an isosceles right triangle.
Sine, cosine and other dependencies establish the relationship between sharp corners and sides of any right triangle. Let us give formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:
As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b in cos form A * c, then we get following formulas for tangent and cotangent:
Trigonometric circle
Graphically, the relationship between the mentioned quantities can be represented as follows:
Circumference, in in this case, represents everything possible values angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the size of the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For α from 180° to 360° (III and IV quarters), sin α can only be a negative value.
Let's try to build trigonometric tables for specific angles and find out the value of the quantities.
Values of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values of trigonometric functions for them are calculated and presented in the form of special tables.
These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.
Angles in tables for trigonometric functions correspond to radian values:
So, it is not difficult to guess that 2π is full circle or 360°.
Properties of trigonometric functions: sine and cosine
In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located at two-dimensional system coordinates
Consider comparison table properties for sine and cosine:
| Sine wave | Cosine |
|---|---|
| y = sinx | y = cos x |
| ODZ [-1; 1] | ODZ [-1; 1] |
| sin x = 0, for x = πk, where k ϵ Z | cos x = 0, for x = π/2 + πk, where k ϵ Z |
| sin x = 1, for x = π/2 + 2πk, where k ϵ Z | cos x = 1, at x = 2πk, where k ϵ Z |
| sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z | cos x = - 1, for x = π + 2πk, where k ϵ Z |
| sin (-x) = - sin x, i.e. the function is odd | cos (-x) = cos x, i.e. the function is even |
| the function is periodic, shortest period- 2π | |
| sin x › 0, with x belonging to the 1st and 2nd quarters or from 0° to 180° (2πk, π + 2πk) | cos x › 0, with x belonging to the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk) |
| sin x ‹ 0, with x belonging to the third and fourth quarters or from 180° to 360° (π + 2πk, 2π + 2πk) | cos x ‹ 0, with x belonging to the 2nd and 3rd quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk) |
| increases in the interval [- π/2 + 2πk, π/2 + 2πk] | increases on the interval [-π + 2πk, 2πk] |
| decreases on intervals [π/2 + 2πk, 3π/2 + 2πk] | decreases on intervals |
| derivative (sin x)’ = cos x | derivative (cos x)’ = - sin x |
Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs coincide, the function is even, in otherwise- odd.
The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:
It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.
Properties of tangentsoids and cotangentsoids
The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values tg and ctg are reciprocals of each other.
- Y = tan x.
- The tangent tends to the values of y at x = π/2 + πk, but never reaches them.
- Least positive period tangents is equal to π.
- Tg (- x) = - tg x, i.e. the function is odd.
- Tg x = 0, for x = πk.
- The function is increasing.
- Tg x › 0, for x ϵ (πk, π/2 + πk).
- Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
- Derivative (tg x)’ = 1/cos 2 x.
Let's consider graphic image cotangentoids below in the text.
Main properties of cotangentoids:
- Y = cot x.
- Unlike the sine and cosine functions, in the tangentoid Y can take on the values of the set of all real numbers.
- The cotangentoid tends to the values of y at x = πk, but never reaches them.
- The smallest positive period of a cotangentoid is π.
- Ctg (- x) = - ctg x, i.e. the function is odd.
- Ctg x = 0, for x = π/2 + πk.
- The function is decreasing.
- Ctg x › 0, for x ϵ (πk, π/2 + πk).
- Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
- Derivative (ctg x)’ = - 1/sin 2 x Correct