One of the areas of mathematics that students struggle with the most 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 using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.
Origins of trigonometry
Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.
Historically, the main object of study in 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 in question 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 in art.
First stage
Initially, people talked about the relationship between angles and sides exclusively using 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 in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract trigonometric equations, which begin in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, and 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 earth's surface, and the surface of any other planet is convex, which means that any surface marking will be in three-dimensional space"arc-shaped".
Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.
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.
The first step is to understand the concepts related to a right 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 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 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, which form a right angle, are called legs. In addition, we must remember that the sum of the angles in a triangle is rectangular system coordinates is 180 degrees.
Definition
Finally, with a firm understanding of the geometric basis, one can turn to the definition of sine, cosine and tangent of an angle.
The sine of an angle is the ratio opposite leg(i.e. the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio adjacent leg to the hypotenuse.
Remember that neither sine nor cosine can be more 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 their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.
Finally, the tangent of an angle is the ratio the opposite side to the adjacent one. Dividing the sine by the cosine will give the same result. Look: according to 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 tangent.
Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.
So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to 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 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 need to know the size of the angle rather than the side.
Many students cannot remember the second formula, which is also very popular when solving school tasks: the sum of one and the square of the tangent of the angle is equal to one divided by the square of the cosine of the angle. Take a closer look: 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, conversion rules and several basic formulas you can at any time withdraw the required more complex formulas on a piece of paper.
Formulas for double angles 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 presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of sine and cosine is added.
There are also formulas associated with arguments in the form double angle. They are completely derived from the previous ones - as a practice, try to get them yourself by taking the alpha angle equal to the beta angle.
Finally, note that double angle formulas can be rearranged to reduce the power 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 the figure, and the size of each side, etc.
The sine theorem states that by dividing the length of each side of a triangle by the opposite angle, we get same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a 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 adjacent angle - 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.
Careless mistakes
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 take a look at the most popular ones.
Firstly, you shouldn't convert fractions to decimals until you get the final result - you can leave the answer as common fraction, unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will be wasting your time on unnecessary mathematical operations. This is especially true for values such as the root of three or the root of two, because they are found in problems at every step. The same goes for 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 you will also demonstrate a complete lack of understanding 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 sine is 30 degrees equal to cosine 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 studying trigonometry because they do not understand its practical 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 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're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: length three sides and sizes three corners. The only difference in the tasks lies in the fact that different input data are given.
You now know how to find sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a ratio, and a ratio is a fraction, main goal trigonometric problem is finding the roots of an ordinary equation or a system of equations. And here regular school mathematics will help you.
Trigonometry is a branch of mathematical science that studies trigonometric functions and their use in geometry. The development of trigonometry began in ancient Greece. During the Middle Ages important contribution Scientists from the Middle East and India contributed to the development of this science.
This article is dedicated to basic concepts and definitions of trigonometry. It discusses the definitions of the basic trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.
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Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of aspect ratios right triangle.
Definitions of trigonometric functions
The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.
Cosine of the angle (cos α) - the ratio of the adjacent leg to the hypotenuse.
Angle tangent (t g α) - the ratio of the opposite side to the adjacent side.
Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.
These definitions are given for acute angle right triangle!
Let's give an illustration.
IN triangle ABC with right angle C sine of angle A equal to the ratio leg BC to hypotenuse AB.
The definitions of sine, cosine, tangent and cotangent allow you to calculate the values of these functions from the known lengths of the sides of the triangle.
Important to remember!
The range of values of sine and cosine is from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of values of tangent and cotangent is the entire number line, that is, these functions can take on any values.
The definitions given above apply to acute angles. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to 0 to 90 degrees. The rotation angle in degrees or radians is expressed by any real number from - ∞ to + ∞.
In this context, we can define sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Let's imagine unit circle centered at the origin of the Cartesian coordinate system.
The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).
Sine (sin) of the rotation angle
The sine of the rotation angle α is the ordinate of point A 1 (x, y). sin α = y
Cosine (cos) of the rotation angle
The cosine of the rotation angle α is the abscissa of point A 1 (x, y). cos α = x
Tangent (tg) of the rotation angle
The tangent of the angle of rotation α is the ratio of the ordinate of point A 1 (x, y) to its abscissa. t g α = y x
Cotangent (ctg) of the rotation angle
The cotangent of the rotation angle α is the ratio of the abscissa of point A 1 (x, y) to its ordinate. c t g α = x y
Sine and cosine are defined for any rotation angle. This is logical, because the abscissa and ordinate of a point after rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is undefined when a point after rotation goes to a point with zero abscissa (0, 1) and (0, - 1). In such cases, the expression for tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with cotangent. The difference is that the cotangent is not defined in cases where the ordinate of a point goes to zero.
Important to remember!
Sine and cosine are defined for any angles α.
Tangent is defined for all angles except α = 90° + 180° k, k ∈ Z (α = π 2 + π k, k ∈ Z)
Cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)
When deciding practical examples do not say "sine of the angle of rotation α". The words “angle of rotation” are simply omitted, implying that it is already clear from the context what is being discussed.
Numbers
What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?
Sine, cosine, tangent, cotangent of a number
Sine, cosine, tangent and cotangent of a number t is a number that is respectively equal to sine, cosine, tangent and cotangent in t radian.
For example, the sine of the number 10 π equal to sine rotation angle of 10 π rad.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's take a closer look at it.
Any real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.
The starting point on the circle is point A with coordinates (1, 0).
Positive number t
Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and will go the way t.
Now that the connection between a number and a point on a circle has been established, we move on to the definition of sine, cosine, tangent and cotangent.
Sine (sin) of t
Sine of a number t- ordinate of a point on the unit circle corresponding to the number t. sin t = y
Cosine (cos) of t
Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x
Tangent (tg) of t
Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t
The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. A point on a circle corresponding to the number t, coincides with the point to which the starting point goes after turning by an angle t radian.
Trigonometric functions of angular and numeric argument
Each value of angle α corresponds to specific value the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except α = 180° k, k ∈ Z (α = π k, k ∈ Z).
We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.
Similarly, we can talk about sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.
Basic functions of trigonometry
Sine, cosine, tangent and cotangent are the basic trigonometric functions.
It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.
Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. Trigonometric definitions sine, cosine, tangent and cotangent are completely consistent with geometric definitions, given using the aspect ratios of a right triangle. Let's show it.
Take a unit circle with center at a rectangular Cartesian system coordinates Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right triangle, angle A 1 O H equal to angle turn α, the length of the leg O H is equal to the abscissa of point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.
In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side to the hypotenuse.
sin α = A 1 H O A 1 = y 1 = y
This means that determining the sine of an acute angle in a right triangle through the aspect ratio is equivalent to determining the sine of the rotation angle α, with alpha lying in the range from 0 to 90 degrees.
Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.
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If we construct a unit circle with center at the origin, and set arbitrary value argument x 0 and count from the axis Ox corner x 0, then this angle on the unit circle corresponds to a certain point A(Fig. 1) and its projection onto the axis Oh there will be a point M. Section length OM equal to absolute value abscissa dots A. This value argument x 0 function value mapped y=cos x 0 like abscissa dots A. Accordingly, point IN(x 0 ;at 0) belongs to the graph of the function at=cos X(Fig. 2). If the point A is to the right of the axis OU, The current sine will be positive, but if to the left it will be negative. But anyway, period A cannot leave the circle. Therefore, the cosine lies in the range from –1 to 1:
–1 = cos x = 1.
Additional rotation at any angle, multiple of 2 p, returns point A to the same place. Therefore the function y = cos xp:
cos( x+ 2p) = cos x.
If we take two values of the argument, equal in absolute value, but opposite in sign, x And - x, find the corresponding points on the circle A x And A -x. As can be seen in Fig. 3 their projection onto the axis Oh is the same point M. That's why
cos(– x) = cos ( x),
those. cosine – even function, f(–x) = f(x).
This means we can explore the properties of the function y=cos X on the segment , and then take into account its parity and periodicity.
At X= 0 point A lies on the axis Oh, its abscissa is 1, and therefore cos 0 = 1. With increasing X dot A moves around the circle up and to the left, its projection, naturally, is only to the left, and at x = p/2 cosine becomes equal to 0. Point A at this moment rises to maximum height, and then continues to move to the left, but already descending. Its abscissa keeps decreasing until it reaches lowest value, equal to –1 at X= p. Thus, on the interval the function at=cos X decreases monotonically from 1 to –1 (Fig. 4, 5).
From the parity of the cosine it follows that on the interval [– p, 0] the function increases monotonically from –1 to 1, taking a zero value at x =–p/2. If you take several periods, you get a wavy curve (Fig. 6).
So the function y=cos x takes zero values at points X= p/2 + kp, Where k – any integer. Maximums equal to 1 are achieved at points X= 2kp, i.e. in steps of 2 p, and minimums equal to –1 at points X= p + 2kp.
Function y = sin x.
On the unit circle corner x 0 corresponds to a dot A(Fig. 7), and its projection onto the axis OU there will be a point N.Z function value y 0 = sin x 0 defined as the ordinate of a point A. Dot IN(corner x 0 ,at 0) belongs to the graph of the function y= sin x(Fig. 8). It is clear that the function y = sin x periodic, its period is 2 p:
sin( x+ 2p) = sin ( x).
For two argument values, X And - , projections of their corresponding points A x And A -x per axis OU located symmetrically relative to the point ABOUT. That's why
sin(– x) = –sin ( x),
those. sine is an odd function, f(– x) = –f( x) (Fig. 9).
If the point A rotate relative to a point ABOUT at an angle p/2 counterclockwise (in other words, if the angle X increase by p/2), then its ordinate in the new position will be equal to the abscissa in the old one. Which means
sin( x+ p/2) = cos x.
Otherwise, the sine is a cosine “late” by p/2, since any cosine value will be “repeated” in the sine when the argument increases by p/2. And to build a sine graph, it is enough to shift the cosine graph by p/2 to the right (Fig. 10). Extremely important property sine is expressed by equality
The geometric meaning of equality can be seen from Fig. 11. Here X - this is half an arc AB, a sin X - half of the corresponding chord. It is obvious that as the points get closer A And IN the length of the chord is increasingly approaching the length of the arc. From the same figure it is easy to derive the inequality
|sin x| x|, true for any X.
Mathematicians call formula (*) remarkable limit. From it, in particular, it follows that sin X» X at small X.
Functions at= tg x, y=ctg X. The other two trigonometric functions, tangent and cotangent, are most easily defined as the ratios of the sine and cosine already known to us:
Like sine and cosine, tangent and cotangent are periodic functions, but their periods are equal p, i.e. they are half the size of sine and cosine. The reason for this is clear: if sine and cosine both change signs, then their ratio will not change.
Since the denominator of the tangent contains a cosine, the tangent is not defined at those points where the cosine is 0 - when X= p/2 +kp. At all other points it increases monotonically. Direct X= p/2 + kp for tangent are vertical asymptotes. At points kp tangent and slope are 0 and 1, respectively (Fig. 12).
The cotangent is not defined where the sine is 0 (when x = kp). At other points it decreases monotonically, and straight lines x = kp – his vertical asymptotes. At points x = p/2 +kp the cotangent becomes 0, and the slope at these points is equal to –1 (Fig. 13).
Parity and periodicity.
A function is called even if f(–x) = f(x). The cosine and secant functions are even, and the sine, tangent, cotangent and cosecant functions are odd:
| sin (–α) = – sin α | tan (–α) = – tan α |
| cos (–α) = cos α | ctg (–α) = – ctg α |
| sec (–α) = sec α | cosec (–α) = – cosec α |
Parity properties follow from the symmetry of points P a and R- a (Fig. 14) relative to the axis X. With such symmetry, the ordinate of the point changes sign (( X;at) goes to ( X; –у)). All functions - periodic, sine, cosine, secant and cosecant have a period of 2 p, and tangent and cotangent - p:
| sin (α + 2 kπ) = sin α | cos(α+2 kπ) = cos α |
| tg(α+ kπ) = tan α | cot(α+ kπ) = cotg α |
| sec (α + 2 kπ) = sec α | cosec(α+2 kπ) = cosec α |
The periodicity of sine and cosine follows from the fact that all points P a+2 kp, Where k= 0, ±1, ±2,…, coincide, and the periodicity of the tangent and cotangent is due to the fact that the points P a+ kp alternately fall into two diametrically opposite points circles that give the same point on the tangent axis.
The main properties of trigonometric functions can be summarized in a table:
| Function | Domain | Multiple meanings | Parity | Areas of monotony ( k= 0, ± 1, ± 2,…) |
| sin x | –Ґ x Ґ | [–1, +1] | odd | increases with x O((4 k – 1) p /2, (4k + 1) p/2), decreases at x O((4 k + 1) p /2, (4k + 3) p/2) |
| cos x | –Ґ x Ґ | [–1, +1] | even | Increases with x O((2 k – 1) p, 2kp), decreases at x O(2 kp, (2k + 1) p) |
| tg x | x № p/2 + p k | (–Ґ , +Ґ ) | odd | increases with x O((2 k – 1) p /2, (2k + 1) p /2) |
| ctg x | x № p k | (–Ґ , +Ґ ) | odd | decreases at x ABOUT ( kp, (k + 1) p) |
| sec x | x № p/2 + p k | (–Ґ , –1] AND [+1, +Ґ ) | even | Increases with x O(2 kp, (2k + 1) p), decreases at x O((2 k– 1) p , 2 kp) |
| cosec x | x № p k | (–Ґ , –1] AND [+1, +Ґ ) | odd | increases with x O((4 k + 1) p /2, (4k + 3) p/2), decreases at x O((4 k – 1) p /2, (4k + 1) p /2) |
Reduction formulas.
According to these formulas, the value of the trigonometric function of the argument a, where p/2 a p , can be reduced to the value of the argument function a , where 0 a p /2, either the same or complementary to it.
| Argument b |  -a |
+ a | p-a | p+ a | + a | + a | 2p-a |
| sin b | cos a | cos a | sin a | –sin a | –cos a | –cos a | –sin a |
| cos b | sin a | –sin a | –cos a | –cos a | –sin a | sin a | cos a |
Therefore, in the tables of trigonometric functions, values are given only for acute angles, and it is enough to limit ourselves, for example, to sine and tangent. The table shows only the most commonly used formulas for sine and cosine. From these it is easy to obtain formulas for tangent and cotangent. When casting a function from an argument of the form kp/2 ± a, where k– an integer, to a function of the argument a:
1) the function name is saved if k even, and changes to "complementary" if k odd;
2) the sign on the right side coincides with the sign of the reducible function at the point kp/2 ± a if angle a is acute.
For example, when casting ctg (a – p/2) we make sure that a – p/2 at 0 a p /2 lies in the fourth quadrant, where the cotangent is negative, and, according to rule 1, we change the name of the function: ctg (a – p/2) = –tg a .
Addition formulas.
Formulas for multiple angles.
These formulas are derived directly from the addition formulas:
sin 2a = 2 sin a cos a ;
cos 2a = cos 2 a – sin 2 a = 2 cos 2 a – 1 = 1 – 2 sin 2 a ;
sin 3a = 3 sin a – 4 sin 3 a ;
cos 3a = 4 cos 3 a – 3 cos a ;
The formula for cos 3a was used by François Viète when solving cubic equation. He was the first to find expressions for cos n a and sin n a , which were later obtained more in a simple way from Moivre's formula.
If in formulas double argument replace a with a /2, they can be converted to half angle formulas:
Universal substitution formulas.
Using these formulas, an expression involving different trigonometric functions of the same argument can be rewritten as rational expression from one function tg (a /2), this can be useful when solving some equations:
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Formulas for converting sums into products and products into sums.
Before the advent of computers, these formulas were used to simplify calculations. Calculations were made using logarithmic tables, and later - a slide rule, because logarithms are best suited for multiplying numbers, so all the original expressions were brought to a form convenient for logarithmization, i.e. to works, for example:
2 sin a sin b = cos ( a–b) – cos ( a+b);
2cos a cos b=cos( a–b) + cos ( a+b);
2 sin a cos b= sin( a–b) + sin ( a+b).
Formulas for the tangent and cotangent functions can be obtained from the above.
Degree reduction formulas.
From the multiple argument formulas the following formulas are derived:
| sin 2 a = (1 – cos 2a)/2; | cos 2 a = (1 + cos 2a )/2; |
| sin 3 a = (3 sin a – sin 3a)/4; | cos 3 a = (3 cos a + cos 3 a )/4. |
Using these formulas trigonometric equations can be reduced to equations of lower degrees. In the same way, we can derive reduction formulas for more high degrees sine and cosine.
| Derivatives and integrals of trigonometric functions | |
| (sin x)` = cos x; | (cos x)` = –sin x; |
| (tg x)` = ; | (ctg x)` = – ; |
| t sin x dx= –cos x + C; | t cos x dx= sin x + C; |
| t tg x dx= –ln|cos x| + C; | t ctg x dx = ln|sin x| + C; |
Each trigonometric function at each point of its domain of definition is continuous and infinitely differentiable. Moreover, the derivatives of trigonometric functions are trigonometric functions, and upon integration, trigonometric functions or their logarithms are also obtained. Integrals of rational combinations of trigonometric functions are always elementary functions.
Representation of trigonometric functions in the form of power series and infinite products.
All trigonometric functions can be expanded in power series. In this case, the functions sin x bcos x are presented in rows. convergent for all values x:
These series can be used to obtain approximate expressions for sin x and cos x at small values x:
at | x| p/2;
at 0 x| p
(B n – Bernoulli numbers).
sin functions x and cos x can be represented in the form of infinite products:
Trigonometric system 1, cos x,sin x, cos 2 x, sin 2 x,¼,cos nx,sin nx, ¼, forms on the segment [– p, p] orthogonal system functions, which makes it possible to represent functions in the form of trigonometric series.
are defined as analytic continuations of the corresponding trigonometric functions of the real argument into the complex plane. Yes, sin z and cos z can be defined using series for sin x and cos x, if instead x put z:
These series converge over the entire plane, so sin z and cos z- entire functions.
Tangent and cotangent are determined by the formulas:
tg functions z and ctg z– meromorphic functions. tg poles z and sec z– simple (1st order) and located at points z = p/2 + pn, poles ctg z and cosec z– also simple and located at points z = p n, n = 0, ±1, ±2,…
All formulas that are valid for trigonometric functions of a real argument are also valid for a complex one. In particular,
sin(– z) = –sin z,
cos(– z) = cos z,
tg(– z) = –tg z,
ctg(– z) = –ctg z,
those. even and odd parity are preserved. Formulas are also saved
sin( z + 2p) = sin z, (z + 2p) = cos z, (z + p) = tg z, (z + p) = ctg z,
those. periodicity is also preserved, and the periods are the same as for functions of a real argument.
Trigonometric functions can be expressed in terms of an exponential function of a purely imaginary argument:
Back, e iz expressed in terms of cos z and sin z according to the formula:
e iz=cos z + i sin z
These formulas are called Euler's formulas. Leonhard Euler developed them in 1743.
Trigonometric functions can also be expressed in terms of hyperbolic functions:
z = –i sh iz, cos z = ch iz, z = –i th iz.
where sh, ch and th are hyperbolic sine, cosine and tangent.
Trigonometric functions of complex argument z = x + iy, Where x And y – real numbers, can be expressed through trigonometric and hyperbolic functions of real arguments, for example:
sin( x + iy) = sin x ch y + i cos x sh y;
cos( x + iy) = cos x ch y + i sin x sh y.
The sine and cosine of a complex argument can take real values, exceeding 1 in absolute value. For example:
If an unknown angle enters an equation as an argument of trigonometric functions, then the equation is called trigonometric. Such equations are so common that their methods the solutions are very detailed and carefully developed. WITH with help various techniques and formulas reduce trigonometric equations to equations of the form f(x)= a, Where f– any of the simplest trigonometric functions: sine, cosine, tangent or cotangent. Then express the argument x this function through its known value A.
Since trigonometric functions are periodic, the same A from the range of values there are infinitely many values of the argument, and the solutions to the equation cannot be written as a single function of A. Therefore, in the domain of definition of each of the main trigonometric functions, a section is selected in which it takes all its values, each only once, and the function inverse to it is found in this section. Such functions are denoted by adding the prefix arc (arc) to the name of the original function, and are called inverse trigonometric functions or simply arc functions.
Inverse trigonometric functions.
For sin X, cos X, tg X and ctg X can be determined inverse functions. They are denoted accordingly by arcsin X(read "arcsine" x"), arcos x, arctan x and arcctg x. By definition, arcsin X there is such a number y, What
sin at = X.
Similarly for other inverse trigonometric functions. But this definition suffers from some inaccuracy.
If you reflect sin X, cos X, tg X and ctg X relative to the bisector of the first and third quadrants coordinate plane, then the functions, due to their periodicity, become ambiguous: the same sine (cosine, tangent, cotangent) corresponds to infinite number corners
To get rid of ambiguity, a section of the curve with a width of p, in this case it is necessary that a one-to-one correspondence be maintained between the argument and the value of the function. Areas near the origin of coordinates are selected. For sine in As a “one-to-one interval” we take the segment [– p/2, p/2], on which the sine monotonically increases from –1 to 1, for the cosine – the segment, for the tangent and cotangent, respectively, the intervals (– p/2, p/2) and (0, p). Each curve on the interval is reflected relative to the bisector and now inverse trigonometric functions can be determined. For example, let the argument value be given x 0 , such that 0 Ј x 0 Ј 1. Then the value of the function y 0 = arcsin x 0 there will be only one meaning at 0 , such that - p/2 Ј at 0 Ј p/2 and x 0 = sin y 0 .
Thus, arcsine is a function of arcsin A, defined on the interval [–1, 1] and equal for each A to such a value, – p/2 a p /2 that sin a = A. It is very convenient to represent it using a unit circle (Fig. 15). When | a| 1 on a circle there are two points with ordinate a, symmetrical about the axis u. One of them corresponds to the angle a= arcsin A, and the other is the corner p - a. WITH taking into account the periodicity of the sine, the solution sin equations x= A is written as follows:
x =(–1)n arcsin a + 2p n,
Where n= 0, ±1, ±2,...
Other simple trigonometric equations can be solved in the same way:
cos x = a, –1 =a= 1;
x =±arcos a + 2p n,
Where P= 0, ±1, ±2,... (Fig. 16);
tg X = a;
x= arctan a + p n,
Where n = 0, ±1, ±2,... (Fig. 17);
ctg X= A;
X= arcctg a + p n,
Where n = 0, ±1, ±2,... (Fig. 18).
Basic properties of inverse trigonometric functions:
arcsin X(Fig. 19): domain of definition – segment [–1, 1]; range – [– p/2, p/2], monotonically increasing function;
arccos X(Fig. 20): domain of definition – segment [–1, 1]; range – ; monotonically decreasing function;
arctg X(Fig. 21): domain of definition – all real numbers; range of values – interval (– p/2, p/2); monotonically increasing function; straight at= –p/2 and y = p /2 – horizontal asymptotes;
arcctg X(Fig. 22): domain of definition – all real numbers; range of values – interval (0, p); monotonically decreasing function; straight y= 0 and y = p– horizontal asymptotes.
For anyone z = x + iy, Where x And y are real numbers, inequalities apply
½| e\e y–e-y| ≤|sin z|≤½( e y +e-y),
½| e y–e-y| ≤|cos z|≤½( e y +e -y),
of which at y® Ґ asymptotic formulas follow (uniformly with respect to x)
|sin z| » 1/2 e |y| ,
|cos z| » 1/2 e |y| .
Trigonometric functions first appeared in connection with research in astronomy and geometry. The ratios of segments in a triangle and a circle, which are essentially trigonometric functions, are found already in the 3rd century. BC e. in the works of mathematicians of Ancient Greece – Euclid, Archimedes, Apollonius of Perga and others, however, these relations were not an independent object of study, so they did not study trigonometric functions as such. They were initially considered as segments and in this form were used by Aristarchus (late 4th - 2nd half of the 3rd centuries BC), Hipparchus (2nd century BC), Menelaus (1st century AD). ) and Ptolemy (2nd century AD) when solving spherical triangles. Ptolemy compiled the first table of chords for acute angles every 30" with an accuracy of 10 -6. This was the first table of sines. As a ratio sin function a is found already in Aryabhata (late 5th century). The functions tg a and ctg a are found in al-Battani (2nd half of the 9th – early 10th centuries) and Abul-Wef (10th century), who also uses sec a and cosec a. Aryabhata already knew the formula (sin 2 a + cos 2 a) = 1, and also sin formulas and cos half angle, with the help of which I built tables of sines for angles every 3°45"; based on known values trigonometric functions for the simplest arguments. Bhaskara (12th century) gave a method for constructing tables in terms of 1 using addition formulas. Formulas for converting the sum and difference of trigonometric functions of various arguments into a product were derived by Regiomontanus (15th century) and J. Napier in connection with the latter’s invention of logarithms (1614). Regiomontan gave a table of sine values in 1". The expansion of trigonometric functions into power series was obtained by I. Newton (1669). In modern form the theory of trigonometric functions was introduced by L. Euler (18th century). He owns their definition for real and complex arguments, the symbolism currently accepted, the establishment of connections with exponential function and orthogonality of the system of sines and cosines.
The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concept of 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 as scary as he is painted,” let’s start from the very beginning and understand the concept of an angle.
Angle concept: radian, degree
Let's look at the picture. The vector has “turned” relative to the point by a certain amount. So the measure of this rotation relative to the initial position will be corner.
What else do you need to know about the concept of angle? Well, of course, angle units!
Angle, in both geometry and trigonometry, can be measured in degrees and radians.
An angle of (one degree) is called central angle in a circle, based on a circular arc equal to part of the circle. Thus, the entire circle consists of “pieces” of circular arcs, or the angle described by the circle is equal.
That is, the figure above shows an angle equal to, that is, this angle rests on a circular arc the size of the circumference.
An angle in radians is the central angle in a circle subtended by a circular arc whose length is equal to the radius of the circle. Well, did you figure it out? If not, then let's figure it out from the 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 radius equal to length arcs). Thus, the arc length is calculated by the formula:
Where is the central angle in radians.
Well, knowing this, can you answer how many radians are contained in the angle described by the circle? Yes, for this you need to remember the formula for circumference. Here she is:
Well, now let’s correlate these two formulas and find that the angle described by the circle is equal. That is, by correlating the value in degrees and radians, we get that. Respectively, . As you can see, unlike "degrees", the word "radian" is omitted, since the unit of measurement is usually clear from the context.
How many radians are there? That's right!
Got it? Then go ahead and fix it:
Having difficulties? Then look answers:
Right triangle: sine, cosine, tangent, cotangent of angle
So, we figured out the concept of an angle. But what is sine, cosine, tangent, and cotangent of an angle? Let's figure it out. To do this, a right 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 the right angle (in our example this is the side); legs are the two remaining sides and (those adjacent to right angle), and, 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 sine, cosine, tangent and cotangent of an angle?
Sine of angle- this is the ratio of the opposite (distant) leg to the hypotenuse.
In our triangle.
Cosine of angle- this is the ratio of the adjacent (close) leg to the hypotenuse.
In our triangle.
Tangent of the angle- this is the ratio of the opposite (distant) side to the adjacent (close).
In our triangle.
Cotangent of angle- this is the ratio of the adjacent (close) leg to the opposite (far).
In our triangle.
These definitions are necessary remember! To make it easier to remember which leg to divide into what, you need to clearly understand that in tangent And cotangent only the legs sit, and the hypotenuse appears only in sinus 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, you need to remember that sine, cosine, tangent and cotangent as the ratios of the sides of a triangle do not depend on the lengths of these sides (at the same 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 understand the definitions, then go ahead and consolidate them!
For the triangle shown in the figure below, we find.
Well, did you get it? Then try it yourself: calculate the same for the angle.
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 will be very useful when studying trigonometry. Therefore, let's look at it in a little more detail.
As you can see, given circle constructed in a Cartesian coordinate system. Circle radius equal to one, while the center of the circle lies at the origin, starting position The radius vector is fixed along the positive direction of the axis (in our example, this is the radius).
Each point on the circle corresponds to two numbers: the axis coordinate and the axis coordinate. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, we need to remember about the considered right triangle. In the figure above, you can see two whole right triangles. Consider a triangle. It is rectangular because it is perpendicular to the axis.
What is the triangle equal to? That's right. In addition, we know that is the radius of the unit circle, which means . Let's substitute this value into our formula for cosine. Here's what happens:
What is the triangle equal to? Well, of course, ! Substitute the radius value into this formula and get:
So, can you tell what coordinates a point belonging to a circle has? Well, no way? What if you realize that and are just numbers? Which coordinate does it correspond to? Well, of course, the coordinates! And what coordinate does it correspond to? That's right, coordinates! Thus, period.
What then are and equal to? That's right, let's use the corresponding definitions of tangent and cotangent and get that, a.
What if the angle is larger? For example, like in this picture:
What has changed in this example? Let's figure it out. To do this, let's turn again to a right triangle. Consider a right triangle: angle (as adjacent to an angle). What are the values 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 - the coordinate; and the values of tangent and cotangent to the corresponding ratios. Thus, these relations apply to any rotation of the radius vector.
It has already been 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 happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain value, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles, and when rotating clockwise - negative.
So, we know that a whole revolution of the radius vector around a circle is or. Is it possible to rotate the radius vector to or to? Well, of course you can! In the first case, therefore, the radius vector will make one full turn and stops in the or position.
In the second case, that is, the radius vector will make three full revolutions and stop at position or.
Thus, from the above examples we can conclude that angles that differ by or (where is any integer) correspond to the same position of the radius vector.
The figure below shows an angle. The same image corresponds to the corner, etc. This list can be continued indefinitely. 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:
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 angle measures. Well, let's start in order: the angle at 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 trigonometric functions at the corresponding points. Try it yourself first, and then check the answers.
Answers:
Does not exist
Does not exist
Does not exist
Does not exist
Thus, we can make the following table:
There is no need to remember all these values. 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 angles in and, given in the table below, must be remembered:
Don't be scared, now we'll show you one example quite simple to remember the corresponding values:
To use this method, it is vital to remember the values of the sine for all three measures of angle (), as well as the value of the tangent of the angle. Knowing these values, it is quite simple to restore the entire table - 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. Cotangent values are transferred in accordance with the arrows indicated in the figure. If you understand this and remember the diagram with the arrows, then it will be enough to remember all the values from the table.
Coordinates of a point 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 get it out general formula to find the coordinates of a point.
For example, here is a circle in front of us:
We are given that the point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of a point obtained by rotating the point by degrees.
As can be seen from the figure, the coordinate of the point corresponds to the length of the segment. The length of the segment corresponds to the coordinate of the center of the circle, that is, it is equal. The length of a segment can be expressed using the definition of cosine:
Then we have that for the point coordinate.
Using the same logic, we find the y coordinate value for the point. Thus,
So, in general view coordinates of points are determined by the formulas:
Coordinates of the center of the circle,
Circle radius,
The rotation angle of the vector radius.
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, let's try out these formulas by practicing finding points on a circle?
1. Find the coordinates of a point on the unit circle obtained by rotating the point on.
2. Find the coordinates of a point on the unit circle obtained by rotating the point on.
3. Find the coordinates of a point on the unit circle obtained by rotating the point on.
4. The point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
5. The point is the center of the circle. The radius of the circle is equal. 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 get good at solving them) and you will learn to find them!
1.
You can notice that. But we know what corresponds to a full revolution of the starting point. Thus, desired point will be in the same position as when turning on. Knowing this, we find the required coordinates of the point:
2. The unit circle is centered at a point, which means we can use simplified formulas:
You can notice that. We know what corresponds to two full revolutions 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:
Sine and cosine are table values. We recall their meanings and get:
Thus, the desired point has coordinates.
3. The unit circle is centered at a point, which means we can use simplified formulas:
You can notice that. Let's depict the example in question in the figure:
The radius makes angles equal to and with the axis. Knowing that the table values of cosine and sine are equal, and having determined that the cosine here takes negative meaning, and the sine is positive, we have:
More details similar examples are understood when studying formulas for reducing trigonometric functions in the topic.
Thus, the desired point has coordinates.
4.
Angle of rotation of the radius of the vector (by condition)
To determine the corresponding signs of sine and cosine, we construct a unit circle and angle:
As you can see, the value, that is, is positive, and the value, that is, is negative. Knowing the tabular values of the corresponding trigonometric functions, we obtain that:
Let's substitute the obtained values into our formula and find the coordinates:
Thus, the desired point has coordinates.
5. To solve this problem, we use formulas in general form, where
Coordinates of the center of the circle (in our example,
Circle radius (by condition)
Angle of rotation of the radius of the vector (by condition).
Let's substitute all the values into the formula and get:
and - table values. Let’s remember and substitute them into the formula:
Thus, the desired point has coordinates.
SUMMARY AND BASIC FORMULAS
The sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.
The cosine of an angle is the ratio of the adjacent (close) leg to the hypotenuse.
The tangent of an angle is the ratio of the opposite (far) side to the adjacent (close) side.
The cotangent of an angle is the ratio of the adjacent (close) side to the opposite (far) side.
Examples:
\(\cos(30^°)=\)\(\frac(\sqrt(3))(2)\)
\(\cos\)\(\frac(π)(3)\) \(=\)\(\frac(1)(2)\)
\(\cos2=-0.416…\)
Argument and meaning
Cosine of an acute angle
Cosine of an acute angle can be determined using a right triangle - it is equal to the ratio of the adjacent leg to the hypotenuse.
Example :
1) Let an angle be given and we need to determine the cosine of this angle.
2) Let us complete any right triangle on this angle.
3) Having measured the required sides, we can calculate the cosine.
Cosine of a number
The number circle allows you to determine the cosine of any number, but usually you find the cosine of numbers somehow related to: \(\frac(π)(2)\) , \(\frac(3π)(4)\) , \(-2π\ ).
For example, for the number \(\frac(π)(6)\) - the cosine will be equal to \(\frac(\sqrt(3))(2)\) . And for the number \(-\)\(\frac(3π)(4)\) it will be equal to \(-\)\(\frac(\sqrt(2))(2)\) (approximately \(-0 ,71\)).
For cosine for other numbers often encountered in practice, see.
The cosine value always lies in the range from \(-1\) to \(1\). In this case, the cosine can be calculated for absolutely any angle and number.
Cosine of any angle
Thanks to number circle You can determine the cosine of not only an acute angle, but also an obtuse, negative, and even greater than \(360°\) (full revolution). How to do this is easier to see once than to hear \(100\) times, so look at the picture.
Now an explanation: suppose we need to determine the cosine of the angle KOA With degree measure in \(150°\). Combining the point ABOUT with the center of the circle, and the side OK– with the \(x\) axis. After this, set aside \(150°\) counterclockwise. Then the ordinate of the point A will show us the cosine of this angle.
If we are interested in an angle with a degree measure, for example, in \(-60°\) (angle KOV), we do the same, but we set \(60°\) clockwise.
And finally, the angle is greater than \(360°\) (angle CBS) - everything is similar to the stupid one, only after going clockwise a full turn, we go to the second circle and “get the lack of degrees”. Specifically, in our case, the angle \(405°\) is plotted as \(360° + 45°\).
It’s easy to guess that to plot an angle, for example, in \(960°\), you need to make two turns (\(360°+360°+240°\)), and for an angle in \(2640°\) - whole seven.
As you could replace, both the cosine of a number and the cosine of an arbitrary angle are defined almost identically. Only the way the point is found on the circle changes.
Cosine signs by quarters
Using the cosine axis (that is, the abscissa axis, highlighted in red in the figure), it is easy to determine the signs of the cosines along the numerical (trigonometric) circle:
Where the values on the axis are from \(0\) to \(1\), the cosine will have a plus sign (I and IV quarters - green area),
- where the values on the axis are from \(0\) to \(-1\), the cosine will have a minus sign (II and III quarters - purple area).
Relation to other trigonometric functions:
- the same angle (or number): main trigonometric identity\(\sin^2x+\cos^2x=1\)- the same angle (or number): by the formula \(1+tg^2x=\)\(\frac(1)(\cos^2x)\)
- and the sine of the same angle (or number): the formula \(ctgx=\)\(\frac(\cos(x))(\sinx)\)
For other most commonly used formulas, see.
Solution of the equation \(\cosx=a\)
The solution to the equation \(\cosx=a\), where \(a\) is a number no greater than \(1\) and no less than \(-1\), i.e. \(a∈[-1;1]\):
\(\cos x=a\) \(⇔\) \(x=±\arccosa+2πk, k∈Z\)
If \(a>1\) or \(a<-1\), то решений у уравнения нет.
Example . Solve the trigonometric equation \(\cosx=\)\(\frac(1)(2)\).Solution:
Let's solve the equation using the number circle. For this:
1) Let's build the axes.
2) Let's construct a circle.
3) On the cosine axis (axis \(y\)) mark the point \(\frac(1)(2)\) .
4) Draw a perpendicular to the cosine axis through this point.
5) Mark the intersection points of the perpendicular and the circle.
6) Let's sign the values of these points: \(\frac(π)(3)\) ,\(-\)\(\frac(π)(3)\) .
7) Let’s write down all the values corresponding to these points using the formula \(x=t+2πk\), \(k∈Z\):
\(x=±\)\(\frac(π)(3)\) \(+2πk\), \(k∈Z\);
Answer: \(x=±\frac(π)(3)+2πk\) \(k∈Z\)
Function \(y=\cos(x)\)
If we plot the angles in radians along the \(x\) axis, and the cosine values corresponding to these angles along the \(y\) axis, we get the following graph:
This graph is called and has the following properties:
The domain of definition is any value of x: \(D(\cos(x))=R\)
- range of values – from \(-1\) to \(1\) inclusive: \(E(\cos(x))=[-1;1]\)
- even: \(\cos(-x)=\cos(x)\)
- periodic with period \(2π\): \(\cos(x+2π)=\cos(x)\)
- points of intersection with coordinate axes:
abscissa axis: \((\)\(\frac(π)(2)\) \(+πn\),\(;0)\), where \(n ϵ Z\)
Y axis: \((0;1)\)
- intervals of constancy of sign:
the function is positive on the intervals: \((-\)\(\frac(π)(2)\) \(+2πn;\) \(\frac(π)(2)\) \(+2πn)\), where \(n ϵ Z\)
the function is negative on the intervals: \((\)\(\frac(π)(2)\) \(+2πn;\)\(\frac(3π)(2)\) \(+2πn)\), where \(n ϵ Z\)
- intervals of increase and decrease:
the function increases on the intervals: \((π+2πn;2π+2πn)\), where \(n ϵ Z\)
the function decreases on the intervals: \((2πn;π+2πn)\), where \(n ϵ Z\)
- maximums and minimums of the function:
the function has a maximum value \(y=1\) at points \(x=2πn\), where \(n ϵ Z\)
the function has a minimum value \(y=-1\) at points \(x=π+2πn\), where \(n ϵ Z\).