Definition of sine and cosine. Sine, cosine, tangent and cotangent - everything you need to know on the exam in mathematics (2020)
Teachers believe that every student should be able to carry out calculations, know trigonometric formulas, but not every teacher explains what sine and cosine are. What is their meaning, where are they used? Why are we talking about triangles, but a circle is drawn in the textbook? Let's try to connect all the facts together.
School subject
The study of trigonometry usually begins in the 7th-8th grade of high school. At this time, students are explained what sine and cosine are, they are offered to solve geometric problems using these functions. Later, more complex formulas and expressions appear that need to be transformed in an algebraic way (double and half angle formulas, power functions), work is carried out with a trigonometric circle.
However, teachers are far from always able to clearly explain the meaning of the concepts used and the applicability of formulas. Therefore, the student often does not see the point in this subject, and memorized information is quickly forgotten. However, it is worth explaining to a high school student once, for example, the connection between function and oscillatory motion, and the logical connection will be remembered for many years, and jokes about the uselessness of the subject will become a thing of the past.
Usage
For the sake of curiosity, let's take a look at various branches of physics. Do you want to determine the range of the projectile? Or are you calculating the frictional force between an object and a certain surface? Swinging the pendulum, watching the rays passing through the glass, calculating the induction? Trigonometric concepts appear in almost any formula. So what are sine and cosine?
Definitions
The sine of the angle is the ratio of the opposite leg to the hypotenuse, the cosine is the ratio of the adjacent leg to the same hypotenuse. There is absolutely nothing complicated here. Perhaps students are usually confused by the values they see in the trigonometric table, because square roots appear there. Yes, it is not very convenient to get decimal fractions from them, but who said that all numbers in mathematics should be equal?
In fact, in the trigonometry problem books, you can find a funny hint: most of the answers here are even and in the worst case contain the root of two or three. The conclusion is simple: if you get a “multi-storey” fraction in your answer, double-check the solution for errors in calculations or in reasoning. And you will most likely find them.
Things to Remember
As with any science, trigonometry has data that needs to be learned.
First, you should remember the numerical values for sines, cosines of a right triangle 0 and 90, as well as 30, 45 and 60 degrees. These indicators are found in nine out of ten school problems. Peeping these values in the textbook, you will waste a lot of time, and there will be no place to look at the test or exam at all.
It should be remembered that the value of both functions cannot exceed one. If anywhere in the calculation you get a value outside of the 0-1 range, stop and re-solve the problem.
The sum of the squares of the sine and cosine is equal to one. If you have already found one of the values, use this formula to find the rest.
Theorems
There are two main theorems in basic trigonometry: sines and cosines.
The first says that the ratio of each side of a triangle to the sine of the opposite angle is the same. The second is that the square of any side can be obtained by adding the squares of the two remaining sides and subtracting their double product, multiplied by the cosine of the angle lying between them.
Thus, if we substitute the value of an angle of 90 degrees into the cosine theorem, we get ... the Pythagorean theorem. Now, if you need to calculate the area of a figure that is not a right-angled triangle, you don't have to worry anymore - the two theorems considered will significantly simplify the solution of the problem.
Targets and goals
Learning trigonometry becomes much easier when you realize one simple fact: all the actions you perform are aimed at achieving just one goal. Any parameters of a triangle can be found if you know the least information about it - it can be the value of one angle and the length of two sides, or, for example, three sides.
To determine the sine, cosine, tangent of any angle, these data are enough, with their help you can easily calculate the area of the figure. Almost always, one of the mentioned values is required as an answer, and you can find them using the same formulas.
Inconsistencies in learning trigonometry
One of the incomprehensible questions that students prefer to avoid is finding a connection between various concepts in trigonometry. It would seem that triangles are used to study the sines and cosines of angles, but for some reason the designations are often found in the figure with a circle. In addition, there is a completely incomprehensible wave-like graph called a sinusoid, which has no external resemblance either to a circle or to triangles.
Moreover, the angles are measured in degrees, then in radians, and the number Pi, written simply as 3.14 (without units of measurement), for some reason appears in the formulas, corresponding to 180 degrees. How does all this relate to each other?
Units
Why is Pi exactly 3.14? Do you remember what this meaning is? This is the number of radii that fit in an arc on half a circle. If the diameter of the circle is 2 centimeters, the circumference is 3.14 * 2, or 6.28.
Second point: you may have noticed the similarity between the words "radian" and "radius". The fact is that one radian is numerically equal to the value of the angle plotted from the center of the circle onto an arc with a length of one radius.
Now let's combine the knowledge gained and understand why the top on the coordinate axis in trigonometry is written "Pi in half", and on the left - "Pi". This is an angular value measured in radians, since a semicircle is 180 degrees, or 3.14 radians. And where there are degrees, there are sines and cosines. The triangle is easy to draw from the desired point, postponing the segments to the center and on the coordinate axis.
Let's look into the future
Trigonometry, studied at school, deals with a rectilinear coordinate system, where, as strange as it may sound, a straight line is a straight line.
But there are also more complex ways of working with space: the sum of the angles of a triangle here will be more than 180 degrees, and a straight line in our view will look like a real arc.
Let's move from words to deeds! Take an apple. Make three cuts with the knife to form a triangle when viewed from above. Take out the resulting apple slice and look at the "ribs" where the rind ends. They are not straight at all. The fruit in your hands can be conditionally called round, and now imagine how complex the formulas must be, with the help of which you can find the area of the cut out piece. But some specialists solve such problems on a daily basis.
Trigonometric functions in life
Have you noticed that the shortest plane route from point A to point B on the surface of our planet has a pronounced arc shape? The reason is simple: the Earth has the shape of a ball, which means that you cannot calculate much with the help of triangles - here you have to use more complex formulas.
The sine / cosine of an acute angle cannot be dispensed with in any space-related matter. It is interesting that a whole variety of factors converge here: trigonometric functions are required when calculating the motion of planets along circles, ellipses and various trajectories of more complex shapes; the process of launching rockets, satellites, shuttles, undocking research vehicles; observation of distant stars and the study of galaxies that humans will not be able to reach in the foreseeable future.
In general, the field for the activity of a person who owns trigonometry is very wide and, apparently, will only expand over time.
Conclusion
Today we learned, or at least repeated what sine and cosine are. These are concepts that you don't need to be afraid of - you just want to, and you will understand their meaning. Remember that trigonometry is not a goal, but only a tool that can be used to meet real human needs: build houses, ensure traffic safety, even explore the vastness of the universe.
Indeed, science itself may seem boring, but as soon as you find in it a way to achieve your own goals, self-realization, the learning process will become interesting, and your personal motivation will increase.
As a homework assignment, try to find ways to apply trigonometric functions to an area of activity that interests you personally. Imagine, turn on your imagination, and then it will probably turn out that new knowledge will be useful to you in the future. And besides, mathematics is useful for the general development of thinking.
| BD |- the length of an arc of a circle centered at a point A.
α
is the angle expressed in radians.
Sinus ( sin α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg | BC | to the length of the hypotenuse | AC |.
Cosine ( cos α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg | AB | to the length of the hypotenuse | AC |.
Accepted designations
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Sine function graph, y = sin x
Cosine function graph, y = cos x
Sine and cosine properties
Periodicity
Functions y = sin x and y = cos x periodic with a period 2 π.
Parity
The sine function is odd. The cosine function is even.
Range of definition and values, extrema, increase, decrease
The sine and cosine functions are continuous in their domain of definition, that is, for all x (see the proof of continuity). Their main properties are presented in the table (n is an integer).
| y = sin x | y = cos x | |
| Domain of definition and continuity | - ∞ < x < + ∞ | - ∞ < x < + ∞ |
| Range of values | -1 ≤ y ≤ 1 | -1 ≤ y ≤ 1 |
| Ascending | ||
| Descending | ||
| Maxima, y = 1 | ||
| Minima, y = - 1 | ||
| Zeros, y = 0 | ||
| Points of intersection with the y-axis, x = 0 | y = 0 | y = 1 |
Basic formulas
Sum of squares of sine and cosine
Sine and cosine formulas for sum and difference
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Formulas for the product of sines and cosines
Sum and Difference Formulas
Expression of sine in terms of cosine
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Cosine expression in terms of sine
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Tangent expression
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For, we have:
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At :
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Table of sines and cosines, tangents and cotangents
This table shows the values of sines and cosines for some values of the argument. 
Expressions using complex variables
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Euler's formula
Expressions in terms of hyperbolic functions
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Derivatives
; ... Derivation of formulas>>>
Derivatives of the nth order:
{ -∞ <
x < +∞ }
Secant, cosecant
Inverse functions
The inverse functions of sine and cosine are inverse sine and inverse cosine, respectively.
Arcsin, arcsin
Arccosine, arccos
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Technical Institutions, "Lan", 2009.
Simplest trigonometric identities
The quotient of dividing the sine of the alpha angle by the cosine of the same angle is equal to the tangent of this angle (Formula 1). See also the proof of the correctness of the transformation of the simplest trigonometric identities.
The quotient of dividing the cosine of the alpha angle by the sine of the same angle is equal to the cotangent of the same angle (Formula 2)
The secant of an angle is equal to one divided by the cosine of the same angle (Formula 3)
The sum of the squares of the sine and cosine of the same angle is equal to one (Formula 4). see also the proof of the sum of squares of cosine and sine.
The sum of the unit and the tangent of an angle is equal to the ratio of the unit to the square of the cosine of this angle (Formula 5)
The unit plus the cotangent of the angle is equal to the quotient of dividing one by the sine square of this angle (Formula 6)
The product of the tangent and the cotangent of the same angle is equal to one (Formula 7).
Convert negative angles of trigonometric functions (even and odd)
In order to get rid of the negative value of the degree measure of the angle when calculating the sine, cosine or tangent, you can use the following trigonometric transformations (identities) based on the principles of evenness or oddness of trigonometric functions.
As seen, cosine and the secant is even function, sine, tangent and cotangent - odd functions.
The sine of a negative angle is equal to the negative sine of that same positive angle (minus sine alpha).
The cosine "minus alpha" will give the same value as the cosine of the angle alpha.
The tangent minus alpha is equal to the minus tangent alpha.
Double angle reduction formulas (sine, cosine, tangent and cotangent of double angle)
If you need to divide an angle in half, or vice versa, go from a double angle to a single angle, you can use the following trigonometric identities:
Double angle conversion (sine of double angle, cosine of double angle and tangent of double angle) to single occurs according to the following rules:
Double angle sine equal to twice the product of sine and cosine of a single angle
Double angle cosine is equal to the difference between the square of the cosine of a single angle and the square of the sine of this angle
Double angle cosine equal to twice the square of the cosine of a single angle minus one
Double angle cosine equal to one minus double sine square of a single angle
Double angle tangent is equal to a fraction, the numerator of which is the double tangent of a single angle, and the denominator is equal to one minus the tangent of the square of a single angle.
Double angle cotangent is equal to a fraction, the numerator of which is the square of the cotangent of a single angle minus one, and the denominator is equal to twice the cotangent of a single angle
Universal trigonometric substitution formulas
The conversion formulas below can be useful when you need to divide the argument of a trigonometric function (sin α, cos α, tan α) by two and reduce the expression to half the angle. From the value of α we obtain α / 2.These formulas are called universal trigonometric substitution formulas... Their value lies in the fact that the trigonometric expression with their help is reduced to the expression of the tangent of half an angle, regardless of which trigonometric functions (sin cos tg ctg) were originally in the expression. After that, the equation with the tangent of half the angle is much easier to solve.
Trigonometric transformations of half an angle
The following are the formulas for the trigonometric conversion of half an angle to an integer value.The value of the argument of the trigonometric function α / 2 is reduced to the value of the argument of the trigonometric function α.
Trigonometric formulas for adding angles
cos (α - β) = cos α cos β + sin α sin β
sin (α + β) = sin α cos β + sin β cos α
sin (α - β) = sin α cos β - sin β cos α
cos (α + β) = cos α cos β - sin α sin β
Tangent and cotangent of the sum of angles alpha and beta can be converted by the following trigonometric function conversion rules:
Tangent of the sum of angles is equal to a fraction, the numerator of which is the sum of the tangent of the first angle and the tangent of the second angle, and the denominator is one minus the product of the tangent of the first angle and the tangent of the second angle.
Angle difference tangent is equal to the fraction, the numerator of which is equal to the difference between the tangent of the reduced angle and the tangent of the subtracted angle, and the denominator is equal to one plus the product of the tangents of these angles.
Cotangent of the sum of angles is equal to a fraction, the numerator of which is equal to the product of the cotangents of these angles plus one, and the denominator is equal to the difference between the cotangent of the second angle and the cotangent of the first angle.
Angle difference cotangent is equal to a fraction, the numerator of which is the product of the cotangents of these angles minus one, and the denominator is equal to the sum of the cotangents of these angles.
These trigonometric identities are convenient to use when you need to calculate, for example, the tangent of 105 degrees (tg 105). If you represent it as tg (45 + 60), then you can use the given identical transformations of the tangent of the sum of the angles, and then simply substitute the tabular values of the tangent 45 and the tangent 60 degrees.
Sum or Difference Conversion Formulas for Trigonometric Functions
Expressions representing a sum of the form sin α + sin β can be transformed using the following formulas:
Triple angle formulas - convert sin3α cos3α tg3α to sinα cosα tgα
Sometimes it is necessary to transform the triple value of the angle so that the angle α becomes the argument of the trigonometric function instead of 3α.In this case, you can use the triple angle transformation formulas (identities):
Transformation Formulas for the Product of Trigonometric Functions
If it becomes necessary to transform the product of sines of different angles of cosines of different angles, or even the product of sine and cosine, then you can use the following trigonometric identities:
In this case, the product of the sine, cosine or tangent functions of different angles will be converted to the sum or difference.
Trigonometric function reduction formulas
You need to use the cast table as follows. In the line, select the function that interests us. The column contains the corner. For example, the sine of the angle (α + 90) at the intersection of the first row and the first column, we find out that sin (α + 90) = cos α.
Trigonometry is a branch of mathematics that studies trigonometric functions and their use in geometry. The development of trigonometry began in the days of ancient Greece. During the Middle Ages, scientists from the Middle East and India made an important contribution 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.
Initially, the definitions of trigonometric functions, the argument of which is an angle, were expressed in terms of the ratios of the sides of a right-angled triangle.
Definitions of trigonometric functions
The sine of the angle (sin α) is the ratio of the leg opposite to this angle to the hypotenuse.
The cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.
The tangent of the angle (t g α) is the ratio of the opposite leg to the adjacent one.
Angle cotangent (c t g α) - the ratio of the adjacent leg to the opposite one.
These definitions are given for an acute angle of a right triangle!
Here's an illustration.
In a triangle ABC with a 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: from -1 to 1. In other words, the sine and cosine take values from -1 to 1. The range of values of the tangent and cotangent is the whole number line, that is, these functions can take any values.
The definitions given above are for sharp corners. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to a frame from 0 to 90 degrees. The angle of rotation in degrees or radians is expressed by any real number from - ∞ to + ∞.
In this context, you can give a definition of sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Imagine the unit circle centered at the origin of the Cartesian coordinate system.
The starting point A with coordinates (1, 0) rotates around the center of the unit circle by some angle α and goes to point A 1. The definition is given through the coordinates of the point A 1 (x, y).
Sine (sin) of the angle of rotation
The sine of the angle of rotation α is the ordinate of point A 1 (x, y). sin α = y
The cosine (cos) of the angle of rotation
The cosine of the angle of rotation α is the abscissa of point A 1 (x, y). cos α = x
Tangent (tg) angle of rotation
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 angle of rotation
The cotangent of the angle of rotation 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 angle of rotation. This is logical, because the abscissa and ordinate of a point after turning can be determined at any angle. The situation is different with tangent and cotangent. The tangent is not defined when the point after turning goes to the point with zero abscissa (0, 1) and (0, - 1). In such cases, the expression for the tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with the cotangent. The difference is that the cotangent is not defined when the ordinate of a point vanishes.
Important to remember!
Sine and cosine are defined for any angle α.
The tangent is defined for all angles except α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z)
The cotangent is defined for all angles except α = 180 ° k, k ∈ Z (α = π k, k ∈ Z)
When solving practical examples, do not say "sine of the angle of rotation α". The words "angle of rotation" are simply omitted, implying that it is clear from the context what it is about.
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 10 π is equal to the sine of the rotation angle of 10 π rad.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's consider it in more detail.
Any real number t a point on the unit circle with a center at the origin of a rectangular Cartesian coordinate system is assigned. Sine, cosine, tangent and cotangent are defined through the coordinates of this point.
The starting point on the circle is point A with coordinates (1, 0).
A positive number t
Negative number t corresponds to the point to which the starting point will go if it moves counterclockwise along the circle and traverses the path t.
Now that the connection between the number and the point on the circle is established, we proceed to the definition of sine, cosine, tangent and cotangent.
The sine (sin) of t
Sine of number t is the ordinate of the point of the unit circle corresponding to the number t. sin t = y
Cosine (cos) of number t
Cosine number t is the abscissa of the point of the unit circle corresponding to the number t. cos t = x
The tangent (tg) of the number t
Tangent of number t- the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t. t g t = y x = sin t cos t
The latter definitions are consistent with and do not contradict the definition given at the beginning of this clause. The point on the circle corresponding to the number t, coincides with the point to which the starting point goes after rotation by an angle t radian.
Trigonometric functions of angular and numeric argument
Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. As well as all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) there corresponds a certain value of the tangent. The cotangent, as mentioned above, is defined for all α, except for 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, you can talk about sine, cosine, tangent and cotangent as functions of a numeric argument. To every real number t corresponds to a specific value of the sine or cosine of a number t... All numbers other than π 2 + π · k, k ∈ Z, correspond to the value of the tangent. The cotangent is similarly defined for all numbers except π k, k ∈ Z.
Basic functions of trigonometry
Sine, cosine, tangent and cotangent are basic trigonometric functions.
It is usually clear from the context which argument of the trigonometric function (angle argument or numeric argument) we are dealing with.
Let's return to the data at the very beginning of the definitions and the angle alpha, lying in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent and cotangent are completely consistent with the geometric definitions given using the aspect ratios of a right-angled triangle. Let's show it.
Take the unit circle centered in a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right-angled triangle, the angle A 1 O H is equal to the angle of rotation 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 to the corner is equal to the ordinate of point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.
According to the definition from geometry, the sine of the angle α is equal to the ratio of the opposite leg 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-angled triangle through the aspect ratio is equivalent to determining the sine of the angle of rotation 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 an angle. Possession of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. That is why trigonometric calculations often cause difficulties for schoolchildren and students. To overcome them, you should get acquainted with trigonometric functions and formulas in more detail.
Concepts in trigonometry
To understand the basic concepts of trigonometry, you must first determine what a right-angled 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 corners is 90 degrees is rectangular. Historically, this figure was often used by people in architecture, navigation, art, astronomy. Accordingly, studying and analyzing the properties of this figure, people came to the calculation of the corresponding ratios of its parameters.
The main categories associated with right-angled triangles are hypotenuse and legs. The hypotenuse is the side of the triangle opposite the right angle. The legs, respectively, are the other 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 at school, but in applied sciences such as astronomy and geodesy, scientists use it. The peculiarity of a triangle in spherical trigonometry is that it always has a sum of angles of more than 180 degrees.
Angles of a triangle
In a right-angled triangle, the sine of an angle is the ratio of the leg opposite to the desired angle to the hypotenuse of the triangle. Accordingly, the cosine is the ratio of the adjacent leg and the hypotenuse. Both of these values are always less than one, since the hypotenuse is always longer than the leg.
The tangent of an angle is a value equal to the ratio of the opposite leg to the adjacent leg of the desired angle, or sine to cosine. Cotangent, in turn, is the ratio of the adjacent leg of the desired angle to the opposite leg. 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 is equal to one. Such a circle is constructed in a Cartesian coordinate system, while the center of the circle coincides with the origin point, and the initial position of the radius vector is determined along the positive direction of the X-axis (abscissa). Each point of the circle has two coordinates: XX and YY, that is, the coordinates of the abscissas and ordinates. Selecting any point on the circle in the XX plane, and dropping the perpendicular from it to the abscissa axis, we obtain a right-angled triangle formed by the radius to the selected point (denoted by the letter C), by the perpendicular drawn to the X-axis (the intersection point is denoted by the letter G), and a 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-angled 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, we define 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. Similarly, sin α = CG.
In addition, knowing these data, you can determine the coordinate of point C on the circle, since cos α = AG, and sin α = CG, which means that point C has the specified coordinates (cos α; sin α). Knowing that the tangent is equal to the ratio of sine to cosine, we can determine that tg α = y / x, and ctg α = x / y. Considering angles in a negative coordinate system, you can calculate that the values of the sine and cosine of some angles may be negative.
Calculations and basic formulas
Values of trigonometric functions
Having considered the essence of trigonometric functions through the unit circle, you can derive the values of these functions for some angles. The values are listed in the table below.
Simplest trigonometric identities
Equations in which an unknown value is present under the sign of a trigonometric function are called trigonometric. Identities with the value sin х = α, k is 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:
- tg x = 0, x = π / 2 + πk.
- tg x = a, x = arctan α + πk.
Identities with the value ctg x = a, where k is any integer:
- ctg x = 0, x = π / 2 + πk.
- ctg x = a, x = arcctg α + πk.
Casting formulas
This category of constant formulas denotes methods that can be used to switch from trigonometric functions of the form to functions of an argument, that is, to bring 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.
The formulas for converting 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 the cosine of an 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. Likewise 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 in terms of their trigonometric functions. Angles are commonly referred to as α and β.
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) = (tan 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 values of the angles α and β.
Double and triple angle formulas
Double and triple angle trigonometric formulas are formulas that relate the functions of the angles 2α and 3α, respectively, with the trigonometric functions of the angle α. Derived from addition formulas:
- sin2α = 2sinα * cosα.
- cos2α = 1 - 2sin ^ 2 α.
- tg2α = 2tgα / (1 - tg ^ 2 α).
- sin3α = 3sinα - 4sin ^ 3 α.
- cos3α = 4cos ^ 3 α - 3cosα.
- tg3α = (3tgα - tan ^ 3 α) / (1-tan ^ 2 α).
The transition from sum to product
Taking into account that 2sinx * cozy = sin (x + y) + sin (x-y), simplifying this formula, we obtain the 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; tgα + tgβ = sin (α + β) / cosα * cosβ; tgα - tgβ = sin (α - β) / cosα * cosβ; cosα + sinα = √2sin (π / 4 ∓ α) = √2cos (π / 4 ± α).
Moving from work to sum
These formulas follow from the identities of the transition of the sum to the 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 the sine and cosine can be expressed in terms of the sine and cosine of the first power of the 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
Universal trigonometric substitution formulas express trigonometric functions in terms of the tangent of a half angle.
- sin x = (2tgx / 2) * (1 + tan ^ 2 x / 2), while x = π + 2πn;
- cos x = (1 - tan ^ 2 x / 2) / (1 + tan ^ 2 x / 2), where x = π + 2πn;
- tan x = (2tgx / 2) / (1 - tan ^ 2 x / 2), where x = π + 2πn;
- ctg x = (1 - tg ^ 2 x / 2) / (2tgx / 2), while x = π + 2πn.
Special cases
Particular cases of the simplest trigonometric equations are given below (k is any integer).
Private for sinus:
| 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 |
The quotients for the cosine are:
| 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 |
Private 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 |
Private 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
Sine theorem
There are two versions of the theorem - simple and extended. Simple theorem of 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, respectively, opposite angles.
Extended sine theorem for an 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 to them. The sides are denoted as a, b, c, and the corresponding opposite angles are α, β, γ. The formula of the tangent theorem is: (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 opposite angles, r is the radius of the inscribed circle, and p is the half-perimeter of the triangle, the following identities are valid:
- ctg A / 2 = (p-a) / r;
- ctg B / 2 = (p-b) / r;
- ctg C / 2 = (p-c) / r.
Applied application
Trigonometry is not only a theoretical science related to mathematical formulas. Its properties, theorems and rules are used in practice by different branches of human activity - astronomy, air 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 you can mathematically express the relationship between the angles and the lengths of the sides in a triangle, and find the required quantities through identities, theorems and rules.
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