In this article we will show you how definitions of sine, cosine, tangent and cotangent of angle and number in trigonometry... Here we will talk about designations, give examples of entries, and give graphic illustrations. In conclusion, let's 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 trace how the idea of sine, cosine, tangent and cotangent is formed in school course mathematics. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right-angled triangle is given. And later trigonometry is studied, which talks about the sine, cosine, tangent and cotangent of the angle of rotation and number. We will give all these definitions, give examples and give the necessary comments.
Acute angle in a right triangle
From the geometry course, the definitions of sine, cosine, tangent and cotangent of an acute angle in a right-angled triangle are known. They are given as the ratio of the sides of a right-angled triangle. Let us give 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.
Acute tangent in a right triangle Is the ratio of the opposite leg to the adjacent one.
Definition.
Acute cotangent in a right triangle- This is the ratio of the adjacent leg to the opposite one.
The designations for sine, cosine, tangent and cotangent are also introduced there - sin, cos, tg and ctg, respectively.
For example, if ABC is a right-angled triangle with a right angle C, then the sine of an 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 known values sine, cosine, tangent, cotangent and length of one of the sides find the lengths of the other sides. For example, if we knew that in a right-angled triangle the leg AC is 3, and the hypotenuse AB is 7, then we could calculate the value of the cosine of an acute angle A by definition: cos∠A = AC / AB = 3/7.
Turning angle
In trigonometry, they begin to look at the angle more widely - they introduce the concept of the angle of rotation. The value of the angle of rotation, in contrast to the acute angle, is not limited by the 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 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 starting point A (1, 0) goes after it is rotated by an angle α around the point O - the origin of the rectangular Cartesian coordinate system and the center of the unit circle.
Definition.
Sine of rotation angleα is the ordinate of point A 1, that is, sinα = y.
Definition.
The cosine of the angle of rotationα is called the abscissa of point A 1, that is, cos α = x.
Definition.
Rotation tangentα is the ratio of the ordinate of point A 1 to its abscissa, that is, tgα = y / x.
Definition.
Rotation cotangentα is the ratio of the abscissa of 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 by an angle α. And tangent and cotangent are not defined for every angle. The tangent is not defined for such angles α, at which the starting 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 a point with a zero ordinate (1, 0) or (−1, 0), and this is the case for angles 180 ° k, k ∈Z (π k is 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 sin, cos, tg and ctg already known to us appear in the definitions, they are also used to denote the sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the designations tan and cot, corresponding to the tangent and cotangent). So the sine of the rotation angle of 30 degrees can be written as sin30 °, the entries 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 designation "rad" is often omitted. For example, the cosine of a rotation angle of three pi rad is usually denoted cos3 · π.
In conclusion of this point, it is worth noting that in a conversation about 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" or, even shorter, "sine of alpha" is usually used. The same applies to cosine, tangent, and cotangent.
Let's also say that the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle are consistent with the definitions just given of sine, cosine, tangent and cotangent of the angle of rotation ranging from 0 to 90 degrees. We will justify this.
Numbers
Definition.
Sine, cosine, tangent and cotangent of a number t is a number equal to 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 an angle in 8 · π is rad is equal to one, therefore, the cosine of the number 8 · π is 1.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is associated with a point of the unit circle centered at the origin of a rectangular coordinate system, and the sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's dwell on this in more detail.
Let's show how the correspondence is established between real numbers and points of a circle:
- the number 0 is associated with the starting point A (1, 0);
- a positive number t is associated with the point of the unit circle, into which we will get, if we move along the circle from the starting point in the counterclockwise direction and travel a path of length t;
- negative number t is associated with the point of the unit circle, into which we will get, if we move along the circle from the starting point in the clockwise direction and travel a path of length | t | ...
Now we turn to the definitions of sine, cosine, tangent and cotangent of the number t. Suppose that the number t corresponds to the point of the circle A 1 (x, y) (for example, the number π / 2; corresponds to the point A 1 (0, 1)).
Definition.
The sine of a number t is called the ordinate of the point of the unit circle corresponding to the number t, that is, sint = y.
Definition.
Cosine number t is called the abscissa of the point of the unit circle corresponding to the number t, that is, cost = x.
Definition.
The tangent of the 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 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.
Note here that the definitions just given are consistent with the definition given at the beginning of this paragraph. Indeed, the point of the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point by an angle of t radians.
It is also worth clarifying this point. Let's say we have sin3. How to understand if the sine of the number 3 or the sine of the rotation angle of 3 radians are we talking about? This is usually clear from the context, otherwise it is most likely irrelevant.
Trigonometric functions of angular and numeric argument
According to the definitions given in the previous paragraph, each angle of rotation α corresponds to a well-defined value of sinα, as well as the value of cosα. In addition, tgα values correspond to all angles of rotation other than 90 ° + 180 ° k, k∈Z (π / 2 + π ) Are the values of ctgα. Therefore sinα, cosα, tgα and ctgα are functions of the angle α. In other words, they are functions of the angular argument.
Similarly, we can talk about the functions sine, cosine, tangent and cotangent of a numeric argument. Indeed, each real number t has a well-defined value sint, as does cost. In addition, tgt values correspond to all numbers other than π / 2 + π k, k∈Z, and ctgt values correspond to numbers π k, k∈Z.
The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.
It is usually clear from the context whether we are dealing with trigonometric functions of an angular argument or a numeric argument. Otherwise, we can consider the independent variable as both a measure of an angle (angular argument) and a numeric argument.
However, the school mainly studies numeric functions, that is, functions whose arguments, like the corresponding function values, are numbers. Therefore, if it comes precisely about functions, it is advisable to consider trigonometric functions as functions of numeric arguments.
Linking definitions from geometry and trigonometry
If we consider the angle of rotation α in the range from 0 to 90 degrees, then the data in the context of trigonometry for determining the sine, cosine, tangent and cotangent of the angle of rotation fully agree with the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let us justify this.
Let us represent the unit circle in the rectangular Cartesian coordinate system Oxy. Let's mark the starting point A (1, 0). Let's rotate it through an angle α ranging from 0 to 90 degrees, we get the point A 1 (x, y). Let us drop the perpendicular A 1 H from point A 1 onto the Ox axis.
It is easy to see that in a right-angled 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 point A 1, that is, | OH | = x, the length of the leg opposite to the angle of the leg A 1 H is equal to the ordinate of 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-angled 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 point A 1, that is, sin α = y. From this it can be seen that determining the sine of an acute angle in a right-angled triangle is equivalent to determining the sine of the angle of rotation α at α from 0 to 90 degrees.
Similarly, it can be shown that the definitions of the cosine, tangent and cotangent of the acute angle α agree with the definitions of the cosine, tangent and cotangent of the angle of rotation α.
Bibliography.
- Geometry. 7-9 grades: textbook. 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.
- A. V. Pogorelov Geometry: Textbook. for 7-9 cl. general education. institutions / A. V. Pogorelov. - 2nd ed. - M .: Education, 2001. - 224 p .: ill. - ISBN 5-09-010803-X.
- Algebra and elementary functions : Tutorial for students of grade 9 high school/ E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences ON Golovin. - 4th ed. Moscow: Education, 1969.
- Algebra: Textbook. for 9 cl. wednesday school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M .: Education, 1990.- 272 p .: ill.- ISBN 5-09-002727-7
- Algebra and the beginning of the analysis: Textbook. for 10-11 cl. general education. institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M .: Education, 2004. - 384 p .: ill. - ISBN 5-09-013651-3.
- A. G. Mordkovich Algebra and the beginning of analysis. Grade 10. At 2 h. Part 1: textbook for educational institutions ( profile level) / A. G. Mordkovich, P. V. Semenov. - 4th ed., Add. - M .: Mnemozina, 2007 .-- 424 p.: Ill. ISBN 978-5-346-00792-0.
- Algebra and the beginning of 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: Textbook. for 10-11 cl. wednesday shk. - 3rd ed. - M .: Education, 1993 .-- 351 p .: ill. - ISBN 5-09-004617-4.
- Gusev V.A., Mordkovich A.G. Mathematics (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.
The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concept of an angle. To understand well these, at first glance, complex concepts(which cause a state of horror in many schoolchildren), and to make sure that "the devil is not so terrible as he is painted", let's start from the very beginning and understand the concept of an angle.
Angle concept: radian, degree
Let's take a look at the picture. The vector "turned" relative to the point by a certain amount. So, the measure of this rotation relative to the initial position will be injection.
What else do you need to know about the concept of an angle? Well, of course, angle units!
Angle, in both geometry and trigonometry, can be measured in degrees and radians.
Angle (one degree) is called the central angle in a circle, resting on a circular arc equal to part of the circle. Thus, the whole circle consists of "pieces" of circular arcs, or the angle described by the circle is equal to.
That is, the picture above shows an angle equal, that is, this angle rests on a circular arc with the size of the circumference.
An angle in radians is the central angle in a circle that rests on a circular arc whose length is equal to the radius of the circle. Well, figured it out? If not, then let's figure it out by drawing.
So, the figure shows an angle equal to a radian, that is, this angle rests on a circular arc, the length of which is equal to the radius of the circle (the length is equal to the length or the radius is equal to the length of the arc). Thus, the arc length is calculated by the formula:
Where is the center angle in radians.
Well, can you, knowing this, answer how many radians the angle described by the circle contains? Yes, for this you need to remember the formula for the circumference. There she is:
Well, now let's relate these two formulas and get that the angle described by the circle is equal. That is, correlating the value in degrees and radians, we get that. Respectively, . As you can see, unlike "degrees", the word "radian" is omitted because the unit is usually clear from context.
How many radians are there? That's right!
Got it? Then fix forward:
Having difficulties? Then look the answers:
Right angle triangle: sine, cosine, tangent, cotangent of an angle
So, we figured out the concept of an angle. But what is sine, cosine, tangent, cotangent of an angle after all? Let's figure it out. For this, a right-angled triangle will help us.
What are the sides of a right triangle called? That's right, hypotenuse and legs: the hypotenuse is the side that lies opposite right angle(in our example, this is the side); the legs are the two remaining sides and (those that are adjacent to the right angle), moreover, if we consider the legs relative to the angle, then the leg is the adjacent leg, and the leg is the opposite. So, now let's answer the question: what are the sine, cosine, tangent and cotangent of an angle?
Sine angle is the ratio of the opposite (distant) leg to the hypotenuse.
In our triangle.
Cosine of an angle is the ratio of the adjacent (close) leg to the hypotenuse.
In our triangle.
Tangent of angle is the ratio of the opposite (distant) leg to the adjacent (close) leg.
In our triangle.
Angle cotangent is the ratio of the adjacent (close) leg to the opposite (distant) leg.
In our triangle.
These definitions are necessary remember! To make it easier to remember which leg to divide into what, you need to clearly realize that in tangent and cotangense only legs sit, and the hypotenuse appears only in sine and cosine... And then you can come up with a chain of associations. For example, this one:
Cosine → touch → touch → adjacent;
Cotangent → touch → touch → adjacent.
First of all, it is necessary to remember that sine, cosine, tangent and cotangent as ratios of the sides of a triangle do not depend on the lengths of these sides (at one angle). Do not believe? Then make sure by looking at the picture:
Consider, for example, the cosine of an angle. By definition, from a triangle:, but we can calculate the cosine of an angle from a triangle:. You see, the lengths of the sides are different, but the value of the cosine of one angle is the same. Thus, the values of sine, cosine, tangent and cotangent depend solely on the magnitude of the angle.
If you figured out the definitions, then go ahead and fix them!
For the triangle shown in the figure below, find.
Well, got it? Then try it yourself: count the same for the corner.
Unit (trigonometric) circle
Understanding the concepts of degrees and radians, we considered a circle with a radius equal to. Such a circle is called single... It comes in very handy when learning trigonometry. Therefore, let's dwell on it in a little more detail.
As you can see, this circle is built in a Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin, the initial position of the radius vector is fixed along the positive direction of the axis (in our example, this is the radius).
Each point of the circle corresponds to two numbers: the coordinate along the axis and the coordinate along the axis. And what are these numbers-coordinates? And in general, what do they have to do with the topic under consideration? To do this, you need to remember about the considered right-angled triangle. In the picture above, you can see two whole right-angled triangles. Consider a triangle. It is rectangular since it is perpendicular to the axis.
What is triangle equal to? That's all right. In addition, we know that - is the radius of the unit circle, and therefore,. Substitute this value into our cosine formula. Here's what happens:
And what is equal to from the triangle? Well, of course, ! Substitute the radius value into this formula and get:
So, can you tell us what are the coordinates of a point belonging to a circle? Well, no way? And if you realize that and are just numbers? What coordinate does it correspond to? Well, of course, the coordinate! And what coordinate does it correspond to? That's right, coordinate! So the point.
And what then are equal to and? That's right, let's use the corresponding definitions of tangent and cotangent and get that, a.
What if the angle is larger? Here, for example, as in this figure:
What has changed in this example? Let's figure it out. To do this, again turn to a right-angled triangle. Consider a right-angled triangle: corner (as adjacent to the corner). What is the value of sine, cosine, tangent and cotangent for an angle? That's right, we adhere to the corresponding definitions of trigonometric functions:
Well, as you can see, the value of the sine of the angle still corresponds to the coordinate; the value of the cosine of the angle - coordinate; and the values of the tangent and cotangent to the corresponding ratios. Thus, these relationships apply to any rotations of the radius vector.
It was already mentioned that the initial position of the radius vector is along the positive direction of the axis. So far we have rotated this vector counterclockwise, but what if we rotated it clockwise? Nothing extraordinary, an angle of a certain magnitude will also turn out, but only it will be negative. Thus, when you rotate the radius vector counterclockwise, you get positive angles, and when rotating clockwise - negative.
So, we know that the whole revolution of the radius vector in a circle is or. Is it possible to rotate the radius vector by or by? Of course you can! In the first case, thus, the radius vector will make one complete revolution and stop at the position or.
In the second case, that is, the radius vector will make three complete revolutions and stop at the position or.
Thus, from the above examples, we can conclude that angles differing by or (where is any integer) correspond to the same position of the radius vector.
The picture below shows the angle. The same image corresponds to the corner, etc. The list goes on and on. All these angles can be written by the general formula or (where is any integer)
Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values are equal to:
Here's a unit circle to help you:
Having difficulties? Then let's figure it out. So, we know that:
From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner corresponds to a point with coordinates, therefore:
Does not exist;
Further, adhering to the same logic, we find out that the corners in correspond to points with coordinates, respectively. Knowing this, it is easy to determine the values of the trigonometric functions at the corresponding points. Try it yourself first, then check the answers.
Answers:
Does not exist
Does not exist
Does not exist
Does not exist
Thus, we can draw up the following table:
It is not necessary to remember all of these meanings. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values of trigonometric functions:
But the values of the trigonometric functions of the angles in and, given in the table below, need to remember:
Do not be afraid, now we will show one of the examples. quite simple memorization of the corresponding values:
To use this method, it is vital to remember the values of the sine for all three measures of the angle (), as well as the value of the tangent of the angle in. Knowing these values, it is quite easy to restore the entire table as a whole - the cosine values are transferred in accordance with the arrows, that is:
Knowing this, you can restore the values for. The numerator "" will match, and the denominator "" will match. The cotangent values are carried over according to the arrows in the figure. If you understand this and remember the diagram with arrows, then it will be enough to remember all the values from the table.
Point coordinates on a circle
Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation?
Well, of course you can! Let's bring general formula for finding the coordinates of a point.
Here, for example, we have such a circle:
We are given that the point is the center of the circle. The radius of the circle is. It is necessary to find the coordinates of the point obtained by turning the point by degrees.
As you can see from the figure, the length of the segment corresponds to the coordinate of the point. The length of the segment corresponds to the coordinate of the center of the circle, that is, it is equal to. The length of a segment can be expressed using the definition of cosine:
Then we have that for the point the coordinate.
Using the same logic, we find the value of the y coordinate for the point. Thus,
So in general view the coordinates of the points are determined by the formulas:
Circle center coordinates,
Circle radius,
The angle of rotation of the radius of the vector.
As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are equal to zero, and the radius is equal to one:
Well, shall we taste these formulas by practicing finding points on a circle?
1. Find the coordinates of a point on the unit circle obtained by turning the point by.
2. Find the coordinates of a point on the unit circle obtained by turning the point by.
3. Find the coordinates of a point on the unit circle obtained by turning the point by.
4. Point is the center of the circle. The radius of the circle is. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
5. Point is the center of the circle. The radius of the circle is. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
Having trouble finding the coordinates of a point on a circle?
Solve these five examples (or understand the solution well) and you will learn how to find them!
1.
You can see that. But we know what corresponds to a complete revolution of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the required coordinates of the point:
2. The circle is unit with a center at a point, which means that we can use simplified formulas:
You can see that. We know that it matches two full turns starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the required coordinates of the point:
Sine and cosine are tabular values. We remember their meanings and get:
Thus, the required point has coordinates.
3. The circle is unit with a center at a point, which means that we can use simplified formulas:
You can see that. Let's depict the considered example in the figure:
The radius makes angles with the axis equal to and. Knowing that the tabular values of the cosine and sine are equal, and determining that the cosine here takes negative meaning, and the sine is positive, we have:
Similar examples are analyzed in more detail when studying the formulas for casting trigonometric functions in the topic.
Thus, the required point has coordinates.
4.
The angle of rotation of the radius of the vector (by condition,)
To determine the corresponding signs of the sine and cosine, we construct the unit circle and angle:
As you can see, the value, that is, positive, and the value, that is, negative. Knowing the tabular values of the corresponding trigonometric functions, we get that:
Substitute the obtained values into our formula and find the coordinates:
Thus, the required point has coordinates.
5. To solve this problem, we will use formulas in general form, where
The coordinates of the center of the circle (in our example,
Circle radius (by condition,)
The angle of rotation of the radius of the vector (by condition,).
Substitute all the values in the formula and get:
and - tabular values. We remember and substitute them in the formula:
Thus, the required point has coordinates.
SUMMARY AND BASIC FORMULAS
The sine of the angle is the ratio of the opposite (far) leg to the hypotenuse.
The cosine of the angle is the ratio of the adjacent (close) leg to the hypotenuse.
The tangent of the angle is the ratio of the opposite (far) leg to the adjacent (close) leg.
The cotangent of an angle is the ratio of the adjacent (close) leg to the opposite (far) leg.
This article contains tables of sines, cosines, tangents and cotangents... First, we give a table of the main values of trigonometric functions, that is, a table of sines, cosines, tangents and cotangents of angles 0, 30, 45, 60, 90, ..., 360 degrees ( 0, π / 6, π / 4, π / 3, π / 2, ..., 2π radian). After that, we will give a table of sines and cosines, as well as a table of tangents and cotangents of V.M. Bradis, and show how to use these tables when finding the values of trigonometric functions.
Page navigation.
Table of sines, cosines, tangents and cotangents for angles 0, 30, 45, 60, 90, ... degrees
Bibliography.
- Algebra: Textbook. for 9 cl. wednesday school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M .: Education, 1990.- 272 p .: ill.- ISBN 5-09-002727-7
- Bashmakov M.I. Algebra and the beginning of analysis: Textbook. for 10-11 cl. wednesday shk. - 3rd ed. - M .: Education, 1993 .-- 351 p .: ill. - ISBN 5-09-004617-4.
- Algebra and the beginning of the analysis: Textbook. for 10-11 cl. general education. institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M .: Education, 2004. - 384 p .: ill. - ISBN 5-09-013651-3.
- Gusev V.A., Mordkovich A.G. Mathematics (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.
- Bradis V.M. Four-digit mathematical tables: For general education. study. institutions. - 2nd ed. - M .: Bustard, 1999. - 96 p .: ill. ISBN 5-7107-2667-2
One of the branches of mathematics with which students cope with the greatest difficulties is trigonometry. It is not surprising: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents by formulas, simplify expressions, and be able to use 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 determining the sine, cosine and tangent of an angle, but first you need to figure out what trigonometry does in general.
Historically, right-angled triangles were the main object of research in this branch of mathematical science. 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 in question on two sides and one corner, or on two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, in astronomy and even in art.
First stage
Initially, people talked about the relationship of angles and sides exclusively on the example of right-angled triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in Everyday life of this section of mathematics.
The study of trigonometry at school today begins with right-angled triangles, after which the knowledge gained is used by students in physics and solving abstract trigonometric equations, work with which begins in high school.
Spherical trigonometry
Later, when science came out on next level 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 "arched" in three-dimensional space.
Take the globe and string. Attach the string to any two points on the globe so that it is taut. Pay attention - it took the shape of an arc. Spherical geometry, which is used in geodesy, astronomy and other theoretical and applied fields, deals with such forms.
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 in this case.
The first step is to understand the concepts related to a right-angled triangle. First, the hypotenuse is the side opposite the 90 degree angle. It 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 the two sides are 3 and 4 centimeters, respectively, the length of the hypotenuse is 5 centimeters. By the way, the ancient Egyptians knew about it about four and a half thousand years ago.
The two remaining sides, which form a right angle, are called legs. In addition, it must be remembered that the sum of the angles in a triangle in a rectangular coordinate system is 180 degrees.
Definition
Finally, with a firm understanding of the geometric base, one can turn to the definition of sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (that is, 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, it will be shorter than the hypotenuse, which means that their ratio will always be less than one. Thus, if you have a sine or cosine with a value greater than 1 in the answer to a problem, look for an error in calculations or reasoning. This answer is definitely wrong.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of the 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 is sine, cosine, tangent and cotangent, and we can do the formulas.
The simplest formulas
In trigonometry, you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.
The first formula that you need to know when starting to learn 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 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 makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, the transformation rules and a few basic formulas, you can at any time you yourself can deduce the required more complex formulas on a sheet of paper.
Double angle formulas and addition of arguments
Two more formulas that you need to learn are related to the values of sine and cosine for the sum and difference of 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, 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 workout, try to get them yourself, taking the alpha angle equal to the angle beta.
Finally, note that the double angle formulas can be transformed to lower the degree of sine, cosine, and 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 hence the area of the figure, and the magnitude of each side, etc.
The sine theorem states that by dividing the length of each side of a 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 the points of the given triangle.
The cosine theorem generalizes the Pythagorean theorem by 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.
Inattentional errors
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to distraction or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.
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 otherwise stated in the condition. Such a transformation cannot be called an error, 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 shortened. 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 are found in problems at every step. The same goes for rounding off "ugly" numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you forget by mistake to subtract the double 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 lack of understanding of the subject. This is worse than a careless mistake.
Third, 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 confuse them, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to start learning trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are the 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 a 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 point of trigonometry boils down to the fact that the unknown parameters of the triangle need to be calculated using the known parameters. There are six of these parameters: length three sides and the magnitude of the three angles. All the difference in tasks is that different inputs are given.
You now know how to find the sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a relation, and a ratio is a fraction, main goal a trigonometric problem is finding the roots of an ordinary equation or a system of equations. And here ordinary school mathematics will help you.
Initially, sine and cosine arose out of the need to calculate quantities in right-angled triangles. It was noticed that if the value of the degree measure of the angles in a right-angled triangle does not change, then the aspect ratio, no matter how much these sides change in length, remains always the same.
This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse, and the cosine is the one adjacent to the hypotenuse.
Cosine and sine theorems
But cosines and sines can be applied not only in right-angled triangles. To find the value of an obtuse or acute angle, the side of any triangle, it is enough to apply the theorem of cosines and sines.
The cosine theorem is quite simple: "The square of the side of a triangle is equal to the sum of the squares of the other two sides minus the double product of these sides by the cosine of the angle between them."
There are two interpretations of the sine theorem: small and extended. According to the small: "In a triangle, the angles are proportional to the opposite sides." This theorem is often extended due to the property of a circle circumscribed about a triangle: "In a triangle, the angles are proportional to the opposite sides, and their ratio is equal to the diameter of the circumscribed circle."
Derivatives
A derivative is a mathematical tool that shows how quickly a function changes relative to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.
When solving problems, you need to know the tabular values of the derivatives of trigonometric functions: sine and cosine. The derivative of the sine is the cosine, and the cosine is the sine, but with a minus sign.
Application in mathematics
Especially often sines and cosines are used when solving right-angled triangles and problems associated with them.
The convenience of sines and cosines is reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking complex shapes and objects into "simple" triangles. Engineers, and often dealing with aspect ratio calculations and degree measures, have spent a lot of time and effort to calculate the cosines and sines of non-tabular angles.
Then Bradis tables came to the rescue, containing thousands of values of sines, cosines, tangents and cotangents of different angles. V Soviet time some teachers made the pages of the Bradis tables by heart to their wards.
Radian - the angular value of the arc, along the length equal to the radius or 57.295779513 ° degrees.
Degree (in geometry) - 1 / 360th of a circle or 1 / 90th of a right angle.
π = 3.141592653589793238462 ... (approximate value of pi).
Cosine table for angles: 0 °, 30 °, 45 °, 60 °, 90 °, 120 °, 135 °, 150 °, 180 °, 210 °, 225 °, 240 °, 270 °, 300 °, 315 °, 330 °, 360 °.
| Angle x (in degrees) | 0° | 30 ° | 45 ° | 60 ° | 90 ° | 120 ° | 135 ° | 150 ° | 180 ° | 210 ° | 225 ° | 240 ° | 270 ° | 300 ° | 315 ° | 330 ° | 360 ° |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Angle x (in radians) | 0 | π / 6 | π / 4 | π / 3 | π / 2 | 2 x π / 3 | 3 x π / 4 | 5 x π / 6 | π | 7 x π / 6 | 5 x π / 4 | 4 x π / 3 | 3 x π / 2 | 5 x π / 3 | 7 x π / 4 | 11 x π / 6 | 2 x π |
| cos x | 1 | √3/2 (0,8660) | √2/2 (0,7071) | 1/2 (0,5) | 0 | -1/2 (-0,5) | -√2/2 (-0,7071) | -√3/2 (-0,8660) | -1 | -√3/2 (-0,8660) | -√2/2 (-0,7071) | -1/2 (-0,5) | 0 | 1/2 (0,5) | √2/2 (0,7071) | √3/2 (0,8660) | 1 |
