Exercise.
Find the value of x at .
Solution.
Finding the value of the function argument at which it is equal to any value means determining at which arguments the value of the sine will be exactly as indicated in the condition.
In this case, we need to find out at what values the sine value will be equal to 1/2. This can be done in several ways.
For example, use , by which to determine at what values of x the sine function will be equal to 1/2.
Another way is to use . Let me remind you that the values of the sines lie on the Oy axis.
The most common way is to use , especially when dealing with values that are standard for this function, such as 1/2.
In all cases, one should not forget about one of the most important properties of the sine - its period.
Let's find the value 1/2 for sine in the table and see what arguments correspond to it. The arguments we are interested in are Pi / 6 and 5Pi / 6.
Let's write down all the roots that satisfy the given equation. To do this, we write down the unknown argument x that interests us and one of the values of the argument obtained from the table, that is, Pi / 6. We write down for it, taking into account the period of the sine, all the values of the argument:
Let's take the second value and follow the same steps as in the previous case:
The complete solution to the original equation will be: 
And 
q can take the value of any integer.
In this article we will show how to give definitions of sine, cosine, tangent and cotangent of an angle and number in trigonometry. Here we will talk about notations, give examples of entries, and give graphic illustrations. In conclusion, let us draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.
Page navigation.
Definition of sine, cosine, tangent and cotangent
Let's see how the idea of sine, cosine, tangent and cotangent is formed in a school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right triangle is given. And later trigonometry is studied, which talks about sine, cosine, tangent and cotangent of the angle of rotation and number. Let us present all these definitions, give examples and give the necessary comments.
Acute angle in a right triangle
From the geometry course we know the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle. They are given as the ratio of the sides of a right triangle. Let us give their formulations.
Definition.
Sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse.
Definition.
Cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.
Definition.
Tangent of an acute angle in a right triangle– this is the ratio of the opposite side to the adjacent side.
Definition.
Cotangent of an acute angle in a right triangle- this is the ratio of the adjacent side to the opposite side.
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 triangle with right angle C, then the sine of the acute angle A is equal to the ratio of the opposite side BC to the hypotenuse AB, that is, sin∠A=BC/AB.
These definitions allow you to calculate the values of sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from the known values of sine, cosine, tangent, cotangent and the length of one of the sides to find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is equal to 3 and the hypotenuse AB is equal to 7, then we could calculate the value of the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7.
Rotation angle
In trigonometry, they begin to look at the angle more broadly - they introduce the concept of angle of rotation. The magnitude of the rotation angle, unlike an acute angle, is not limited to 0 to 90 degrees; the rotation angle 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 given not of an acute angle, but of an angle of arbitrary size - the angle of rotation. They are given through the x and y coordinates of the point A 1, to which the so-called starting point A(1, 0) goes after its rotation by an angle α around the point O - the beginning 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.
Cosine of the rotation angleα is called the abscissa of point A 1, that is, cosα=x.
Definition.
Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tanα=y/x.
Definition.
Cotangent of the rotation angleα is the ratio of the abscissa of point A 1 to its ordinate, that is, ctgα=x/y.
Sine and cosine are defined for any angle α, since we can always determine the abscissa and ordinate of the point, which is obtained by rotating the starting point by angle α. But tangent and cotangent are not defined for any angle. The tangent is not defined for angles α at which the starting point goes to a point with zero abscissa (0, 1) or (0, −1), and this occurs 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 angles α at which the starting point goes to the point with the zero ordinate (1, 0) or (−1, 0), and this occurs for angles 180° k, k ∈Z (π·k rad).
So, sine and cosine are defined for any rotation angles, tangent is defined for all angles except 90°+180°k, k∈Z (π/2+πk rad), and cotangent is defined for all angles except 180° ·k , k∈Z (π·k rad).
The definitions include the designations already known to us sin, cos, tg and ctg, they are also used to designate sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the designations tan and cotcorresponding to tangent and cotangent). So the sine of a 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 when talking 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 rotation angle alpha,” the phrase “sine of the alpha angle” or even shorter, “sine alpha,” is usually used. The same applies to cosine, tangent, and cotangent.
We will 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 for sine, cosine, tangent and cotangent of an 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 the sine, cosine, tangent and cotangent of the rotation angle in t radians, respectively.
For example, the cosine of the number 8·π by definition is a number equal to the cosine of the angle of 8·π rad. And the cosine of an angle of 8·π rad is equal to one, therefore, the cosine of the number 8·π is equal to 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 on the unit circle with the center at the origin of the rectangular coordinate system, and sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's look at this in more detail.
Let us show how a correspondence is established between real numbers and points on a circle:
- the number 0 is assigned the starting point A(1, 0);
- the positive number t is associated with a point on the unit circle, which we will get to if we move along the circle from the starting point in a counterclockwise direction and walk a path of length t;
- the negative number t is associated with a point on the unit circle, which we will get to if we move along the circle from the starting point in a clockwise direction and walk a path of length |t| .
Now we move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point on the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1)).
Definition.
Sine of the number t is the ordinate of the point on the unit circle corresponding to the number t, that is, sint=y.
Definition.
Cosine of the number t is called the abscissa of the point of the unit circle corresponding to the number t, that is, cost=x.
Definition.
Tangent of the number t is the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of a number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost.
Definition.
Cotangent of the number t is the ratio of the abscissa to the ordinate of a point on the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is this: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t: ctgt=cost/sint.
Here we note that the definitions just given are consistent with the definition given at the beginning of this paragraph. Indeed, the point on 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 still worth clarifying this point. Let's say we have the entry sin3. How can we understand whether we are talking about the sine of the number 3 or the sine of the rotation angle of 3 radians? This is usually clear from the context, otherwise it is likely not of fundamental importance.
Trigonometric functions of angular and numeric argument
According to the definitions given in the previous paragraph, each angle of rotation α corresponds to a very specific value sinα, as well as the value cosα. In addition, all rotation angles other than 90°+180°k, k∈Z (π/2+πk rad) correspond to tgα values, and values other than 180°k, k∈Z (πk rad ) – values of ctgα . Therefore sinα, cosα, tanα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.
We can speak similarly about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a very specific value sint, as well as cost. In addition, all numbers other than π/2+π·k, k∈Z correspond to values tgt, and numbers π·k, k∈Z - values ctgt.
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 numerical argument. Otherwise, we can think of the independent variable as both a measure of the angle (angular argument) and a numeric argument.
However, at school we mainly study numerical functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking specifically about functions, then it is advisable to consider trigonometric functions as functions of numerical arguments.
Relationship between definitions from geometry and trigonometry
If we consider the rotation angle α ranging from 0 to 90 degrees, then the definitions of sine, cosine, tangent and cotangent of the rotation angle in the context of trigonometry are fully consistent 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's justify this.
Let us depict the unit circle in the rectangular Cartesian coordinate system Oxy. Let's mark the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get point A 1 (x, y). Let us drop the perpendicular A 1 H from point A 1 to the Ox axis.
It is easy to see that in a right triangle, the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of point A 1, that is, |A 1 H|=y, and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y. And by definition from trigonometry, the sine of the rotation angle α is equal to the ordinate of point A 1, that is, sinα=y. This shows that determining the sine of an acute angle in a right triangle is equivalent to determining the sine of the rotation angle α when α is from 0 to 90 degrees.
Similarly, it can be shown that the definitions of cosine, tangent and cotangent of an acute angle α are consistent with the definitions of cosine, tangent and cotangent of the rotation angle α.
Bibliography.
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- Pogorelov A.V. Geometry: Textbook. for 7-9 grades. general education institutions / A. V. Pogorelov. - 2nd ed. - M.: Education, 2001. - 224 p.: ill. - ISBN 5-09-010803-X.
- Algebra and elementary functions: Textbook for students of 9th grade of secondary school / E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences O. N. Golovin. - 4th ed. M.: Education, 1969.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
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How to find sine?
Studying geometry helps develop thinking. This subject is necessarily included in school training. In everyday life, knowledge of this subject can be useful - for example, when planning an apartment.
From the history
The geometry course also includes trigonometry, which studies trigonometric functions. In trigonometry we study sines, cosines, tangents and cotangents of angles.
But for now, let's start with the simplest thing - sine. Let's take a closer look at the very first concept - the sine of an angle in geometry. What is sine and how to find it?
The concept of “sine angle” and sinusoids
The sine of an angle is the ratio of the values of the opposite side and the hypotenuse of a right triangle. This is a direct trigonometric function, which is written as “sin (x)”, where (x) is the angle of the triangle.
On the graph, the sine of an angle is indicated by a sine wave with its own characteristics. A sine wave looks like a continuous wavy line that lies within certain limits on the coordinate plane. The function is odd, therefore it is symmetrical about 0 on the coordinate plane (it comes out from the origin of the coordinates).
The domain of definition of this function lies in the range from -1 to +1 on the Cartesian coordinate system. The period of the sine angle function is 2 Pi. This means that every 2 Pi the pattern repeats and the sine wave goes through a full cycle.
Sine wave equation
- sin x = a/c
- where a is the leg opposite to the angle of the triangle
- c - hypotenuse of a right triangle
Properties of the sine of an angle
- sin(x) = - sin(x). This feature demonstrates that the function is symmetrical, and if the values x and (-x) are plotted on the coordinate system in both directions, then the ordinates of these points will be opposite. They will be at an equal distance from each other.
- Another feature of this function is that the graph of the function increases on the segment [- P/2 + 2 Pn]; [P/2 + 2Pn], where n is any integer. A decrease in the graph of the sine of the angle will be observed on the segment: [P/2 + 2Pn]; [3P/2 + 2Pn].
- sin(x) > 0 when x is in the range (2Пn, П + 2Пn)
- (x)< 0, когда х находится в диапазоне (-П+2Пn, 2Пn)
The values of the sines of the angle are determined using special tables. Such tables have been created to facilitate the process of calculating complex formulas and equations. It is easy to use and contains not only the values of the sin(x) function, but also the values of other functions.
Moreover, a table of standard values of these functions is included in the compulsory memory study, like a multiplication table. This is especially true for classes with a physical and mathematical bias. In the table you can see the values of the main angles used in trigonometry: 0, 15, 30, 45, 60, 75, 90, 120, 135, 150, 180, 270 and 360 degrees.
There is also a table defining the values of trigonometric functions of non-standard angles. Using different tables, you can easily calculate the sine, cosine, tangent and cotangent of some angles.
Equations are made with trigonometric functions. Solving these equations is easy if you know simple trigonometric identities and reductions of functions, for example, such as sin (P/2 + x) = cos (x) and others. A separate table has also been compiled for such reductions.
How to find the sine of an angle
When the task is to find the sine of an angle, and according to the condition we only have the cosine, tangent, or cotangent of the angle, we can easily calculate what we need using trigonometric identities.
- sin 2 x + cos 2 x = 1
From this equation, we can find both sine and cosine, depending on which value is unknown. We get a trigonometric equation with one unknown:
- sin 2 x = 1 - cos 2 x
- sin x = ± √ 1 - cos 2 x
- cot 2 x + 1 = 1 / sin 2 x
From this equation you can find the value of the sine, knowing the value of the cotangent of the angle. To simplify, replace sin 2 x = y and you have a simple equation. For example, the cotangent value is 1, then:
- 1 + 1 = 1/y
- 2 = 1/y
- 2у = 1
- y = 1/2
Now we perform the reverse replacement of the player:
- sin 2 x = ½
- sin x = 1 / √2
Since we took the cotangent value for the standard angle (45 0), the obtained values can be checked in the table.
If you are given a tangent value and need to find the sine, another trigonometric identity will help:
- tg x * ctg x = 1
It follows that:
- cot x = 1 / tan x
In order to find the sine of a non-standard angle, for example, 240 0, you need to use angle reduction formulas. We know that π corresponds to 180 0. Thus, we express our equality using standard angles by expansion.
- 240 0 = 180 0 + 60 0
We need to find the following: sin (180 0 + 60 0). Trigonometry has reduction formulas that are useful in this case. This is the formula:
- sin (π + x) = - sin (x)
Thus, the sine of an angle of 240 degrees is equal to:
- sin (180 0 + 60 0) = - sin (60 0) = - √3/2
In our case, x = 60, and P, respectively, 180 degrees. We found the value (-√3/2) from the table of values of functions of standard angles.
In this way, non-standard angles can be expanded, for example: 210 = 180 + 30.
Lesson objectives:
The main didactic goal: to consider all possible ways to solve this equation.
Educational: learning new techniques for solving trigonometric equations using the example of a seminar lesson given in a creative situation.
Developmental: formation of general techniques for solving trigonometric equations; improving students' mental operations; development of skills in oral monologue mathematical speech when presenting the solution to a trigonometric equation.
Educators: develop independence and creativity; contribute to the development in schoolchildren of the desire and need to generalize the facts being studied.
Questions for preparation and further discussion at the seminar.
All students are divided into groups (2-4 people) depending on the total number of students and their individual abilities and desires. They independently determine for themselves the topic for preparation and presentation at the lesson-seminar. One person from the group speaks, and the rest of the students take part in additions and corrections of errors, if necessary.
Organizing time.
Students are informed:
Lesson topic:
“Different ways to solve the trigonometric equation sin x - cos x = 1
Form: lesson - seminar.
Epigraph for the lesson:
“A major scientific discovery provides a solution to a major problem, but in the solution of any problem there is a grain of discovery. The problem you solve may be modest, but if it challenges your curiosity and forces you to be inventive, and if you solve it on your own, then you can experience the mental tension that leads to discovery and enjoy the joy of victory.”
(D. Polya)
Lesson objectives:
a) consider the possibility of solving the same equation in different ways;
b) become familiar with various general techniques for solving trigonometric equations;
c) studying new material (introduction of an auxiliary angle, universal substitution).
Seminar plan
- Reducing the equation to a homogeneous equation with respect to sine and cosine.
- Factoring the left side of the equation.
- Introduction of an auxiliary angle.
- Converting the difference (or sum) of trigonometric functions into a product.
- Reduction to a quadratic equation for one of the functions.
- Square both sides of the equation.
- Expression of all functions through tg x (universal substitution).
- Graphical solution of the equation.
1. The floor is given to the first participant.
Reducing the equation sin x - cos x = 1 to a homogeneous equation with respect to sine and cosine.
Let's expand the left-hand side according to the double argument formulas, and replace the right-hand side with a trigonometric unit, using the basic trigonometric identity:
2 sin cos - cos + sin = sin + cos;
2 sin cos - cos =0 ;
cos 
= 0;
The product is equal to zero if at least one of the factors is equal to zero, and the others do not lose their meaning, therefore it follows 
cos =0 ; = 
= 0 - homogeneous equation of the first degree. We divide both sides of the equation by cos. (cos 0, since if cos = 0, then sin - 0 = 0 sin = 0, and this contradicts the trigonometric identity sin + cos = 1).
Answer:
2. The floor is given to the second participant.
Factoring the left side of the equation sin x - cos x = 1.
sin x – (1+ cos x) = 1; we use the formulas 1+ cos x = 2, 
we get 
;
further similar:
the product is equal to zero if at least one of the factors is equal to zero, and the others do not lose their meaning, therefore it follows
cos =0 ; = = 0 - homogeneous equation of the first degree. We divide both sides of the equation by cos. (cos 0, since if cos = 0, then sin - 0 = 0 sin = 0, and this contradicts the trigonometric identity sin + cos = 1)
We get tg -1 = 0 ; tg = 1 ; =
Answer:
3. The floor is given to the third participant.
Solving the equation sin x - cos x = 1 by introducing an auxiliary angle.
Consider the equation sin x - cos x = 1. Multiply and divide each term on the left side
equations for . We get 
and put it outside the bracket on the left side of the equation. We get 
; Let's divide both sides of the equation by and use the tabular values of trigonometric functions. We get 
; Let's apply the sine difference formula. 
;
It is easy to establish (using a trigonometric circle) that the resulting solution splits into two cases: 
; 
Answer:
4. The floor is given to the fourth participant.
Solving the equation sin x - cos x = 1 by converting the difference (or sum) of trigonometric functions into a product.
Let's write the equation in the form 
using the reduction formula 
. Applying the formula for the difference of two sines, we get
;
Answer:
5. The floor is given to the fifth participant.
Solving the equation sin x - cos x = 1 by reducing it to a quadratic equation for one of the functions.
Consider the basic trigonometric identity 
, which follows 
Let's substitute the resulting expression into this equation.
sin x - cos x = 1 
,
Let's square both sides of the resulting equation:
During the solution process, both sides of the equation were squared, which could lead to the appearance of extraneous solutions, so verification is necessary. Let's do it.
The resulting solutions are equivalent to combining three solutions: 
The first and second solutions coincide with those previously obtained, therefore they are not extraneous. It remains to check the third solution 
Let's substitute.
Left side:
Right side: 1.
We got: therefore, - an outside decision.
Answer:
6. The floor is given to the sixth participant.
Square both sides of the equation sin x - cos x = 1.
Consider the equation sin x - cos x = 1. Let's square both sides of this equation.
;
Using the basic trigonometric identity and the double angle sine formula, we obtain ; sin 2x = 0 ; 
. does not make sense, i.e. or .
It is necessary to check whether they are solutions to this equation. Let's substitute these solutions into the left and right sides of the equation.
Left side: .
Right side: 1.
We got 1=1. This means that this is the solution to this equation.
Answer:
8. The floor is given to the eighth participant.
Let's consider a graphical solution to the equation sin x - cos x = 1.
Let us write the equation under consideration in the form sin x = 1 + cos x.
Let us construct graphs of functions corresponding to the left and right sides of the equation in the Oxy coordinate system. The abscissas of the intersection points of the graphs are solutions to this equation.
y = sin x – graph: sinusoid.
y = cos x +1 – graph: cosine wave y = cos x, shifted by 1 upward along the Oy axis. The abscissas of the intersection points are solutions to this equation.
Answer:
Lesson summary.
List of used literature:
- Tatarchenkova S.S. Lesson as a pedagogical phenomenon - St. Petersburg: Karo, 2005
- Vygodsky N.V. Handbook of elementary mathematics.-M.: Nauka, 1975.
- Vilenkin N.Ya. and others. Behind the pages of a mathematics textbook: Arithmetic. Algebra. Geometry: A book for students in grades 10-11 - M.: Education, 1996.
- Gnedenko B.V. Essays on the history of mathematics in Russia - M.: OGIZ, 1946.
- Depman I.Ya. and others. Behind the pages of a mathematics textbook - M.: Education, 1999.
- Dorofeev G.V. and others. Mathematics: for those entering universities - M.: Bustard, 2000.
- Mathematics: Large encyclopedic dictionary. – M.: TSB, 1998.
- Mordkovich A.G. etc. Schoolchildren's Handbook on Mathematics. 10-11 grades Algebra and the beginnings of analysis. – M.: Aquarium, 1997.
- 300 competitive problems in mathematics. – M.: Rolf, 2000.
- 3600 problems on algebra and principles of analysis. – M.: Bustard, 1999.
- School curriculum in tables and formulas. Large universal reference book. – M.: Bustard, 1999.
- Torosyan V.G. History of education and pedagogical thought: textbook. for university students. - M.: Publishing house VLADOS-PRESS, 2006.- 351 p.
- Krylova N.B. Pedagogical, psychological and moral support as a space for personal changes in a child and an adult. // Class teacher. - 2000. - No. 3. –P.92-103.
Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and orientation by the stars. These calculations related to spherical trigonometry, while in the school course they study the ratio of sides and angles of a plane triangle.
Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationships between the sides and angles of triangles.
During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.
Basic quantities of trigonometry
The basic trigonometric functions of a numeric argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.
The formulas for calculating the values of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants are equal in all directions,” since the proof is given using the example of an isosceles right triangle.
Sine, cosine and other relationships establish the relationship between the acute angles and sides of any right triangle. Let us present formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:
As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, we obtain the following formulas for tangent and cotangent:
Trigonometric circle
Graphically, the relationship between the mentioned quantities can be represented as follows:
The circle, in this case, represents all possible values of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For α from 180° to 360° (III and IV quarters), sin α can only be a negative value.
Let's try to build trigonometric tables for specific angles and find out the meaning of the quantities.
Values of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values of trigonometric functions for them are calculated and presented in the form of special tables.
These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.
Angles in tables for trigonometric functions correspond to radian values:
So, it is not difficult to guess that 2π is a complete circle or 360°.
Properties of trigonometric functions: sine and cosine
In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.
Consider the comparative table of properties for sine and cosine:
| Sine wave | Cosine |
|---|---|
| y = sinx | y = cos x |
| ODZ [-1; 1] | ODZ [-1; 1] |
| sin x = 0, for x = πk, where k ϵ Z | cos x = 0, for x = π/2 + πk, where k ϵ Z |
| sin x = 1, for x = π/2 + 2πk, where k ϵ Z | cos x = 1, at x = 2πk, where k ϵ Z |
| sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z | cos x = - 1, for x = π + 2πk, where k ϵ Z |
| sin (-x) = - sin x, i.e. the function is odd | cos (-x) = cos x, i.e. the function is even |
| the function is periodic, the smallest period is 2π | |
| sin x › 0, with x belonging to the 1st and 2nd quarters or from 0° to 180° (2πk, π + 2πk) | cos x › 0, with x belonging to the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk) |
| sin x ‹ 0, with x belonging to the third and fourth quarters or from 180° to 360° (π + 2πk, 2π + 2πk) | cos x ‹ 0, with x belonging to the 2nd and 3rd quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk) |
| increases in the interval [- π/2 + 2πk, π/2 + 2πk] | increases on the interval [-π + 2πk, 2πk] |
| decreases on intervals [π/2 + 2πk, 3π/2 + 2πk] | decreases on intervals |
| derivative (sin x)’ = cos x | derivative (cos x)’ = - sin x |
Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs coincide, the function is even, otherwise it is odd.
The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:
It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.
Properties of tangentsoids and cotangentsoids
The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values tg and ctg are reciprocals of each other.
- Y = tan x.
- The tangent tends to the values of y at x = π/2 + πk, but never reaches them.
- The smallest positive period of the tangentoid is π.
- Tg (- x) = - tg x, i.e. the function is odd.
- Tg x = 0, for x = πk.
- The function is increasing.
- Tg x › 0, for x ϵ (πk, π/2 + πk).
- Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
- Derivative (tg x)’ = 1/cos 2 x.
Consider the graphic image of the cotangentoid below in the text.
Main properties of cotangentoids:
- Y = cot x.
- Unlike the sine and cosine functions, in the tangentoid Y can take on the values of the set of all real numbers.
- The cotangentoid tends to the values of y at x = πk, but never reaches them.
- The smallest positive period of a cotangentoid is π.
- Ctg (- x) = - ctg x, i.e. the function is odd.
- Ctg x = 0, for x = π/2 + πk.
- The function is decreasing.
- Ctg x › 0, for x ϵ (πk, π/2 + πk).
- Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
- Derivative (ctg x)’ = - 1/sin 2 x Correct
