In this article, we will show how definitions of sine, cosine, tangent and cotangent of angle and number in trigonometry. Here we will talk about notation, give examples of records, give graphic illustrations. In conclusion, we draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.
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Definition of sine, cosine, tangent and cotangent
Let's follow how the concept of sine, cosine, tangent and cotangent is formed in the school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right triangle is given. And later trigonometry is studied, which refers to the sine, cosine, tangent and cotangent of the angle of rotation and the number. We give all these definitions, give examples and give the necessary comments.
Acute angle in a right triangle
From the course of geometry, the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle are known. They are given as the ratio of the sides of a right triangle. We present their formulations.
Definition.
Sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse.
Definition.
Cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.
Definition.
Tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.
Definition.
Cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite leg.
The notation of sine, cosine, tangent and cotangent is also introduced there - sin, cos, tg and ctg, respectively.
For example, if ABC is a right triangle with a right angle C, then the sine of the acute angle A is equal to the ratio of the opposite leg BC to the hypotenuse AB, that is, sin∠A=BC/AB.
These definitions allow you to calculate the values of the sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from the known values of the sine, cosine, tangent, cotangent and the length of one of the sides, find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is 3 and the hypotenuse AB is 7 , then we could calculate the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7 .
Angle of rotation
In trigonometry, they begin to look at the angle more widely - they introduce the concept of angle of rotation. The angle of rotation, unlike an acute angle, is not limited by frames from 0 to 90 degrees, the angle of rotation in degrees (and in radians) can be expressed by any real number from −∞ to +∞.
In this light, the definitions of the sine, cosine, tangent and cotangent are no longer an acute angle, but an angle of arbitrary magnitude - the angle of rotation. They are given through the x and y coordinates of the point A 1 , into which the so-called initial point A(1, 0) passes after it rotates through an angle α around the point O - the beginning of a rectangular Cartesian coordinate system and the center of the unit circle.
Definition.
Sine of rotation angleα is the ordinate of the point A 1 , that is, sinα=y .
Definition.
cosine of the angle of rotationα is called the abscissa of the point A 1 , that is, cosα=x .
Definition.
Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tgα=y/x .
Definition.
The cotangent of the angle of rotationα is the ratio of the abscissa of the point A 1 to its ordinate, that is, ctgα=x/y .
The sine and cosine are defined for any angle α , since we can always determine the abscissa and ordinate of a point, which is obtained by rotating the starting point through the angle α . And tangent and cotangent are not defined for any angle. The tangent is not defined for such angles α at which the initial point goes to a point with zero abscissa (0, 1) or (0, −1) , and this takes place at angles 90°+180° k , k∈Z (π /2+π k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for such angles α at which the starting point goes to to a point with zero ordinate (1, 0) or (−1, 0) , and this is the case for angles 180° k , k ∈Z (π k rad).
So, the sine and cosine are defined for any rotation angles, the tangent is defined for all angles except 90°+180° k , k∈Z (π/2+π k rad), and the cotangent is for all angles except 180° ·k , k∈Z (π·k rad).
The notations already known to us appear in the definitions sin, cos, tg and ctg, they are also used to denote the sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the notation tan and cotcorresponding to tangent and cotangent). So the sine of the rotation angle of 30 degrees can be written as sin30°, the records tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α . Recall that when writing the radian measure of an angle, the notation "rad" is often omitted. For example, the cosine of a rotation angle of three pi rads is usually denoted cos3 π .
In conclusion of this paragraph, it is worth noting that in talking about the sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase "sine of the angle of rotation alpha", the phrase "sine of the angle of alpha" is usually used, or even shorter - "sine of alpha". The same applies to cosine, and tangent, and cotangent.
Let's also say that the definitions of the sine, cosine, tangent, and cotangent of an acute angle in a right triangle are consistent with the definitions just given for the sine, cosine, tangent, and cotangent of a rotation angle ranging from 0 to 90 degrees. We will substantiate this.
Numbers
Definition.
Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the angle of rotation in t radians, respectively.
For example, the cosine of 8 π is, by definition, a number equal to the cosine of an angle of 8 π rad. And the cosine of the angle is 8 π rad equal to one, therefore, the cosine of the number 8 π is equal to 1 .
There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is assigned to a point on the unit circle centered at the origin of the rectangular coordinate system, and the sine, cosine, tangent, and cotangent are defined in terms of the coordinates of this point. Let's dwell on this in more detail.
Let us show how the correspondence between real numbers and points of the circle is established:
- the number 0 is assigned the starting point A(1, 0) ;
- a positive number t is associated with a point on the unit circle, which we will get to if we move around the circle from the starting point in a counterclockwise direction and go through a path of length t;
- negative number t corresponds to a point on the unit circle, which we will reach if we move around the circle from the starting point in a clockwise direction and go through a path of length |t| .
Now let's move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point of the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).
Definition.
The sine of a number t is the ordinate of the unit circle point corresponding to the number t , that is, sint=y .
Definition.
The cosine of a number t is called the abscissa of the point of the unit circle corresponding to the number t , that is, cost=x .
Definition.
Tangent of a number t is the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of the number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost .
Definition.
Cotangent of a number t is the ratio of the abscissa to the ordinate of the point of the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is as follows: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t : ctgt=cost/sint .
Here we note that the definitions just given agree with the definition given at the beginning of this subsection. Indeed, the point of the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point through an angle of t radians.
It is also worth clarifying this point. Let's say we have a sin3 entry. How to understand whether the sine of the number 3 or the sine of the rotation angle of 3 radians is in question? This is usually clear from the context, otherwise it probably doesn't matter.
Trigonometric functions of angular and numerical argument
According to the definitions given in the previous paragraph, each rotation angle α corresponds to a well-defined value sin α , as well as the value cos α . In addition, all rotation angles other than 90°+180° k , k∈Z (π/2+π k rad) correspond to the values tgα , and other than 180° k , k∈Z (π k rad ) are the values of ctgα . Therefore sinα, cosα, tgα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.
Similarly, we can talk about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a well-defined value of sint , as well as cost . In addition, all numbers other than π/2+π·k , k∈Z correspond to the values tgt , and the numbers π·k , k∈Z correspond to the values ctgt .
The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.
It is usually clear from the context that we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can consider the independent variable as both a measure of the angle (the angle argument) and a numeric argument.
However, the school mainly studies numeric functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking about functions, then it is advisable to consider trigonometric functions functions of numeric arguments.
Connection of definitions from geometry and trigonometry
If we consider the angle of rotation α from 0 to 90 degrees, then the data in the context of trigonometry of the definition of the sine, cosine, tangent and cotangent of the angle of rotation are fully consistent with the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's substantiate this.
Draw in a rectangle Cartesian system Oxy coordinates unit circle. Note the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get the point A 1 (x, y) . Let's drop the perpendicular A 1 H from the point A 1 to the Ox axis.
It is easy to see that in a right triangle the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of the point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of the point A 1 , that is, |A 1 H|=y , and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y . And by definition from trigonometry, the sine of the angle of rotation α is equal to the ordinate of the point A 1, that is, sinα=y. This shows that the definition of the sine of an acute angle in a right triangle is equivalent to the definition of the sine of the angle of rotation α for α from 0 to 90 degrees.
Similarly, it can be shown that the definitions of the cosine, tangent, and cotangent of an acute angle α are consistent with the definitions of the cosine, tangent, and cotangent of the angle of rotation α.
Bibliography.
- Geometry. 7-9 grades: studies. for general education institutions / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev and others]. - 20th ed. M.: Education, 2010. - 384 p.: ill. - ISBN 978-5-09-023915-8.
- Pogorelov A.V. Geometry: Proc. for 7-9 cells. general education institutions / A. V. Pogorelov. - 2nd ed. - M.: Enlightenment, 2001. - 224 p.: ill. - ISBN 5-09-010803-X.
- Algebra and elementary functions: Tutorial for students of the 9th grade of secondary school / E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences O. N. Golovin. - 4th ed. Moscow: Education, 1969.
- Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
- Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
- Mordkovich A. G. Algebra and the beginnings of analysis. Grade 10. At 2 h. Part 1: a textbook for educational institutions ( profile level)/ A. G. Mordkovich, P. V. Semenov. - 4th ed., add. - M.: Mnemosyne, 2007. - 424 p.: ill. ISBN 978-5-346-00792-0.
- Algebra and start mathematical analysis. Grade 10: textbook. for general education institutions: basic and profile. levels /[Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; ed. A. B. Zhizhchenko. - 3rd ed. - I .: Education, 2010. - 368 p.: Ill. - ISBN 978-5-09-022771-1.
- Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
Allows you to establish a number of characteristic results - properties of sine, cosine, tangent and cotangent. In this article, we will look at three main properties. The first of them indicates the signs of the sine, cosine, tangent and cotangent of the angle α, depending on which coordinate quarter angle is α. Next, we consider the periodicity property, which establishes the invariance of the values of the sine, cosine, tangent and cotangent of the angle α when this angle changes by an integer number of revolutions. The third property expresses the relationship between the values of the sine, cosine, tangent and cotangent of opposite angles α and −α.
If you are interested in the properties of the functions of sine, cosine, tangent and cotangent, then they can be studied in the corresponding section of the article.
Page navigation.
Signs of sine, cosine, tangent and cotangent in quarters
Below in this paragraph the phrase "angle I, II, III and IV of the coordinate quarter" will be found. Let's explain what these corners are.
Let's take a unit circle, mark the starting point A(1, 0) on it, and rotate it around the point O by an angle α, while we assume that we get to the point A 1 (x, y) .
They say that angle α is the angle I , II , III , IV of the coordinate quarter if point A 1 lies in I, II, III, IV quarters, respectively; if the angle α is such that the point A 1 lies on any of the coordinate lines Ox or Oy , then this angle does not belong to any of the four quarters.
For clarity, we present a graphic illustration. The drawings below show rotation angles of 30 , -210 , 585 and -45 degrees, which are the angles I , II , III and IV of the coordinate quarters, respectively.
corners 0, ±90, ±180, ±270, ±360, … degrees do not belong to any of the coordinate quarters.
Now let's figure out which signs have the values of sine, cosine, tangent and cotangent of the rotation angle α, depending on which quarter angle is α.
For sine and cosine, this is easy to do.
By definition, the sine of the angle α is the ordinate of the point A 1 . It is obvious that in the I and II coordinate quarters it is positive, and in the III and IV quarters it is negative. Thus, the sine of the angle α has a plus sign in the I and II quarters, and a minus sign in the III and VI quarters.
In turn, the cosine of the angle α is the abscissa of the point A 1 . In I and IV quarters it is positive, and in II and III quarters it is negative. Therefore, the values of the cosine of the angle α in the I and IV quarters are positive, and in the II and III quarters they are negative.
To determine the signs by quarters of tangent and cotangent, you need to remember their definitions: tangent is the ratio of the ordinate of point A 1 to the abscissa, and cotangent is the ratio of the abscissa of point A 1 to the ordinate. Then from number division rules with the same and different signs it follows that the tangent and cotangent have a plus sign when the abscissa and ordinate signs of point A 1 are the same, and have a minus sign when the abscissa and ordinate signs of point A 1 are different. Therefore, the tangent and cotangent of the angle have a + sign in the I and III coordinate quarters, and a minus sign in the II and IV quarters.
Indeed, for example, in the first quarter, both the abscissa x and the ordinate y of point A 1 are positive, then both the quotient x/y and the quotient y/x are positive, therefore, the tangent and cotangent have + signs. And in the second quarter of the abscissa, x is negative, and the y-ordinate is positive, so both x / y and y / x are negative, whence the tangent and cotangent have a minus sign.
Let's move on to the next property of sine, cosine, tangent and cotangent.
Periodicity property
Now we will analyze, perhaps, the most obvious property of the sine, cosine, tangent and cotangent of an angle. It consists in the following: when the angle changes by an integer number of full revolutions, the values of the sine, cosine, tangent and cotangent of this angle do not change.
This is understandable: when the angle changes by an integer number of revolutions, we will always get from the starting point A to point A 1 on the unit circle, therefore, the values of sine, cosine, tangent and cotangent remain unchanged, since the coordinates of the point A 1 are unchanged.
Using formulas, the considered property of sine, cosine, tangent and cotangent can be written as follows: sin(α+2 π z)=sinα , cos(α+2 π z)=cosα , tg(α+2 π z)=tgα , ctg(α+2 π z)=ctgα , where α is the rotation angle in radians, z is any , absolute value which indicates the number of full revolutions by which the angle α changes, and the sign of the number z indicates the direction of rotation.
If the rotation angle α is given in degrees, then these formulas will be rewritten as sin(α+360° z)=sinα , cos(α+360° z)=cosα , tg(α+360° z)=tgα , ctg(α+360° z)=ctgα .
Let us give examples of the use of this property. For instance, 
, because 
, a 
. Here is another example: or .
This property, together with reduction formulas, is very often used when calculating the values of the sine, cosine, tangent and cotangent of "large" angles.
The considered property of sine, cosine, tangent and cotangent is sometimes called the periodicity property.
Properties of sines, cosines, tangents and cotangents of opposite angles
Let А 1 be the point obtained as a result of the rotation of the initial point А(1, 0) around the point O by the angle α , and the point А 2 is the result of the rotation of the point А by the angle −α opposite to the angle α .
The property of sines, cosines, tangents and cotangents of opposite angles is based on a fairly obvious fact: the points A 1 and A 2 mentioned above either coincide (at) or are located symmetrically about the axis Ox. That is, if point A 1 has coordinates (x, y) , then point A 2 will have coordinates (x, −y) . From here, according to the definitions of sine, cosine, tangent and cotangent, we write down the equalities and.
Comparing them, we arrive at relations between sines, cosines, tangents and cotangents of opposite angles α and −α of the form .
This is the considered property in the form of formulas.
Let us give examples of the use of this property. For example, the equalities and 
.
It remains only to note that the property of sines, cosines, tangents and cotangents of opposite angles, like the previous property, is often used when calculating the values of sine, cosine, tangent and cotangent, and allows you to completely get away from negative angles.
Bibliography.
- Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
- Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
- Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
We begin our study of trigonometry with a right triangle. Let's define what the sine and cosine are, as well as the tangent and cotangent of an acute angle. These are the basics of trigonometry.
Recall that right angle is an angle equal to 90 degrees. In other words, half of the unfolded corner.
Sharp corner- less than 90 degrees.
Obtuse angle- greater than 90 degrees. In relation to such an angle, "blunt" is not an insult, but a mathematical term :-)
Let's draw a right triangle. A right angle is usually denoted . Note that the side opposite the corner is denoted by the same letter, only small. So, the side lying opposite the angle A is denoted.
An angle is denoted by the corresponding Greek letter.
Hypotenuse A right triangle is the side opposite the right angle.
Legs- sides opposite sharp corners.
The leg opposite the corner is called opposite(relative to angle). The other leg, which lies on one side of the corner, is called adjacent.
Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:
Tangent acute angle in a right triangle - the ratio of the opposite leg to the adjacent:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent leg to the opposite (or, equivalently, the ratio of cosine to sine):
Pay attention to the basic ratios for sine, cosine, tangent and cotangent, which are given below. They will be useful to us in solving problems.
Let's prove some of them.
Okay, we have given definitions and written formulas. But why do we need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is.
We know the relationship between parties right triangle. This is the Pythagorean theorem: .
It turns out that knowing two angles in a triangle, you can find the third one. Knowing two sides in a right triangle, you can find the third. So, for angles - their ratio, for sides - their own. But what to do if in a right triangle one angle (except for a right one) and one side are known, but you need to find other sides?
This is what people faced in the past, making maps of the area and the starry sky. After all, it is not always possible to directly measure all the sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric functions of the angle- give the ratio between parties and corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of sine, cosine, tangent and cotangent values for "good" angles from to.
Notice the two red dashes in the table. For the corresponding values of the angles, the tangent and cotangent do not exist.
Let's analyze several problems in trigonometry from the Bank of FIPI tasks.
1. In a triangle, the angle is , . Find .
The problem is solved in four seconds.
Insofar as , .
2. In a triangle, the angle is , , . Find .
Let's find by the Pythagorean theorem.
Problem solved.
Often in problems there are triangles with angles and or with angles and . Memorize the basic ratios for them by heart!
For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.
A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.
We considered problems for solving right triangles - that is, for finding unknown sides or angles. But that's not all! V USE options in mathematics, there are many problems where the sine, cosine, tangent or cotangent of the outer angle of the triangle appears. More on this in the next article.
Where the tasks for solving a right-angled triangle were considered, I promised to present a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which leg belongs to the hypotenuse (adjacent or opposite). I decided not to put it off indefinitely, necessary material below, please see
The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember very well that the leg refers to the hypotenuse, but which one- forget and confused. The price of a mistake, as you know in the exam, is a lost score.
The information that I will present directly to mathematics has nothing to do. It is associated with figurative thinking, and with the methods of verbal-logical connection. That's right, I myself, once and for all remembereddefinition data. If you still forget them, then with the help of the presented techniques it is always easy to remember.
Let me remind you the definitions of sine and cosine in a right triangle:
Cosine acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:
Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:
So, what associations does the word cosine evoke in you?
Probably everyone has their ownRemember the link:
Thus, you will immediately have an expression in your memory -
«… ratio of ADJACENT leg to hypotenuse».
The problem with the definition of cosine is solved.
If you need to remember the definition of the sine in a right triangle, then remembering the definition of the cosine, you can easily establish that the sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse. After all, there are only two legs, if the adjacent leg is “occupied” by the cosine, then only the opposite side remains for the sine.
What about tangent and cotangent? Same confusion. Students know that this is the ratio of legs, but the problem is to remember which one refers to which - either opposite to adjacent, or vice versa.
Definitions:
Tangent an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one:
Cotangent acute angle in a right triangle is the ratio of the adjacent leg to the opposite:
How to remember? There are two ways. One also uses a verbal-logical connection, the other - a mathematical one.
MATHEMATICAL METHOD
There is such a definition - the tangent of an acute angle is the ratio of the sine of an angle to its cosine:
* Remembering the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one.
Likewise.The cotangent of an acute angle is the ratio of the cosine of an angle to its sine:
So! Remembering these formulas, you can always determine that:
- the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent
- the cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite one.
VERBAL-LOGICAL METHOD
About tangent. Remember the link:
That is, if you need to remember the definition of the tangent, using this logical connection, you can easily remember what it is
"... the ratio of the opposite leg to the adjacent"
If it comes to cotangent, then remembering the definition of tangent, you can easily voice the definition of cotangent -
"... the ratio of the adjacent leg to the opposite"
There is an interesting technique for memorizing tangent and cotangent on the site " Mathematical tandem " , look.
METHOD UNIVERSAL
You can just grind.But as practice shows, thanks to verbal-logical connections, a person remembers information for a long time, and not only mathematical.
I hope the material was useful to you.
Sincerely, Alexander Krutitskikh
P.S: I would be grateful if you tell about the site in social networks.
One of the branches of mathematics with which schoolchildren cope with the greatest difficulties is trigonometry. No wonder: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to apply trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to deduce complex logical chains.
Origins of trigonometry
Acquaintance with this science should begin with the definition of the sine, cosine and tangent of the angle, but first you need to figure out what trigonometry does in general.
Historically, the main object of study of this section mathematical science were right triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values of all parameters of the figure under consideration using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy, and even art.
First stage
Initially, people talked about the relationship of angles and sides exclusively on the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in Everyday life this branch of mathematics.
The study of trigonometry at school today begins with right triangles, after which the knowledge gained is used by students in physics and solving abstract problems. trigonometric equations, work with which begins in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence, at least because the earth's surface, and the surface of any other planet, is convex, which means that any surface marking will be "arc-shaped" in three-dimensional space.
Take the globe and thread. Attach the thread to any two points on the globe so that it is taut. Pay attention - it has acquired the shape of an arc. It is with such forms that spherical geometry, which is used in geodesy, astronomy, and other theoretical and applied fields, deals.
Right triangle
Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.
First of all, it is necessary to understand the concepts related to right triangle. First, the hypotenuse is the side opposite the 90 degree angle. She is the longest. We remember that, according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.
For example, if two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.
The two remaining sides that form a right angle are called legs. In addition, we must remember that the sum of the angles in a triangle in rectangular system coordinate is 180 degrees.
Definition
Finally, with a solid understanding of the geometric base, we can turn to the definition of the sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.
Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means that their ratio will always be less than one. Thus, if you get a sine or cosine with a value greater than 1 in the answer to the problem, look for an error in calculations or reasoning. This answer is clearly wrong.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. The same result will give the division of the sine by the cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, after which we divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same ratio as in the definition of tangent.
The cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing the unit by the tangent.
So, we have considered the definitions of what sine, cosine, tangent and cotangent are, and we can deal with formulas.
The simplest formulas
In trigonometry, one cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? And this is exactly what is required when solving problems.
The first formula that you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you want to know the value of the angle, not the side.
Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: after all, this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation does trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, the conversion rules and a few basic formulas, you can at any time derive the required more complex formulas on a piece of paper.
Double angle formulas and addition of arguments
Two more formulas that you need to learn are related to the values \u200b\u200bof the sine and cosine for the sum and difference of the angles. They are shown in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second case, the pairwise product of the sine and cosine is added.
There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself, taking the angle of alpha equal to the angle of beta.
Finally, note that the double angle formulas can be converted to lower the degree of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of \u200b\u200bthe figure, and the size of each side, etc.
The sine theorem states that as a result of dividing the length of each of the sides of the triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all points of the given triangle.
The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product, multiplied by the double cosine of the angle adjacent to them - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.
Mistakes due to inattention
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's get acquainted with the most popular of them.
Firstly, you should not convert ordinary fractions to decimals until the final result is obtained - you can leave the answer in the form common fraction unless the condition states otherwise. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the task, new roots may appear, which, according to the author's idea, should be reduced. In this case, you will waste time on unnecessary mathematical operations. This is especially true for values such as the root of three or two, because they occur in tasks at every step. The same applies to rounding "ugly" numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but also demonstrate a complete misunderstanding of the subject. This is worse than a careless mistake.
Thirdly, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to mix them up, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to start studying trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on the surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
Finally
So you are sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole essence of trigonometry boils down to the fact that unknown parameters must be calculated from the known parameters of the triangle. There are six parameters in total: the lengths of three sides and the magnitudes of three angles. The whole difference in the tasks lies in the fact that different input data are given.
How to find the sine, cosine, tangent based on the known lengths of the legs or the hypotenuse, you now know. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of the trigonometric problem is to find the roots of an ordinary equation or a system of equations. And here you will be helped by ordinary school mathematics.