Signs of trigonometric functions. Basic properties of trigonometric functions: even, odd, periodicity

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If you are already familiar with trigonometric circle , and just want to refresh your memory individual elements, or you are completely impatient, then here it is:

Here we will analyze everything in detail step by step.

The trigonometric circle is not a luxury, but a necessity

Trigonometry Many people associate it with an impenetrable thicket. Suddenly there are so many meanings trigonometric functions, so many formulas... But it didn’t work out at first, and... off and on... complete misunderstanding...

It is very important not to give up values ​​of trigonometric functions, - they say, you can always look at the spur with a table of values.

If you constantly look at a table with values trigonometric formulas, let's get rid of this habit!

He will help us out! You will work with it several times, and then it will pop up in your head. What is he better tables? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!

For example, say while looking at standard table values ​​of trigonometric formulas , why equal to sine, say 300 degrees, or -45.

No way?.. you can, of course, connect reduction formulas... And looking at the trigonometric circle, you can easily answer such questions. And you will soon know how!

And when deciding trigonometric equations and inequalities without the trigonometric circle - nowhere at all.

Introduction to the trigonometric circle

Let's go in order.

First, let's write out this series of numbers:

And now this:

And finally this:

Of course, it is clear that, in fact, in first place is , in second place is , and in last place is . That is, we will be more interested in the chain.

But how beautiful it turned out! If something happens, we will restore this “miracle ladder.”

And why do we need it?

This chain is the main values ​​of sine and cosine in the first quarter.

Let's draw in rectangular system coordinates is a circle of unit radius (that is, we take any radius in length, and declare its length to be unit).

From the “0-Start” beam we lay down the corners in the direction of the arrow (see figure).

We get the corresponding points on the circle. So, if we project the points onto each of the axes, then we will get exactly the values ​​​​from the above chain.

Why is this, you ask?

Let's not analyze everything. Let's consider principle, which will allow you to cope with other, similar situations.

Triangle AOB is rectangular and contains . And we know that opposite the angle b lies a leg half the size of the hypotenuse (we have the hypotenuse = the radius of the circle, that is, 1).

This means AB= (and therefore OM=). And according to the Pythagorean theorem

I hope something is already becoming clear?

So point B will correspond to the value, and point M will correspond to the value

Same with the other values ​​of the first quarter.

As you understand, the familiar axis (ox) will be cosine axis, and the axis (oy) – axis of sines . Later.

To the left of zero along the cosine axis (below zero along the sine axis) there will, of course, be negative values.

So, here it is, the ALMIGHTY, without whom there is nowhere in trigonometry.

But we’ll talk about how to use the trigonometric circle in.

Trigonometry, as a science, originated in the Ancient East. First trigonometric ratios were developed by astronomers to create an accurate calendar and navigate by the stars. These calculations related to spherical trigonometry, while in school course study the ratios 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 Ancient East to Greece. But the main discoveries of trigonometry are the merit of husbands 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

Basic trigonometric functions numeric argument– these 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 get following formulas for tangent and cotangent:

Trigonometric circle

Graphically, the relationship between the mentioned quantities can be represented as follows:

Circumference, in in this case, represents everything possible values angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the size of 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 value 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’s not difficult to guess that 2π is full 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 comparison table properties for sine and cosine:

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.

1. Y = tan x.
2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
3. Least positive period tangents is equal to π.
4. Tg (- x) = - tg x, i.e. the function is odd.
5. Tg x = 0, for x = πk.
6. The function is increasing.
7. Tg x › 0, for x ϵ (πk, π/2 + πk).
8. Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
9. Derivative (tg x)’ = 1/cos 2 ⁡x.

Let's consider graphic image cotangentoids below in the text.

Main properties of cotangentoids:

1. Y = cot x.
2. Unlike the sine and cosine functions, in the tangentoid Y can take on the values ​​of the set of all real numbers.
3. The cotangentoid tends to the values ​​of y at x = πk, but never reaches them.
4. The smallest positive period of a cotangentoid is π.
5. Ctg (- x) = - ctg x, i.e. the function is odd.
6. Ctg x = 0, for x = π/2 + πk.
7. The function is decreasing.
8. Ctg x › 0, for x ϵ (πk, π/2 + πk).
9. Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
10. Derivative (ctg x)’ = - 1/sin 2 ⁡x Correct

Sinus numbers A is called the ordinate of the point representing this number on number circle. Sine of angle in A radian is called the sine of a number A.

Sinus- number function x. Her domain

Sine range- segment from -1 before 1 , since any number of this segment on the ordinate axis is a projection of any point on the circle, but no point outside this segment is a projection of any of these points.

Sine period

Sine sign:

1. sine equal to zero at , where n- any integer;

2. sine is positive at , where n- any integer;

3. sine is negative when

Where n- any integer.

Sinus- function odd x And -x, then their ordinates - sines - will also turn out to be opposite. That is for anyone x.

1. Sine increases on segments , Where n- any integer.

2. Sine decreases on the segment , Where n- any integer.

At ;

at .

Cosine

Cosine numbers A The abscissa of the point representing this number on the number circle is called. Cosine of the angle in A radian is called the cosine of a number A.

Cosine- function of number. Her domain- the set of all numbers, since for any number you can find the ordinate of the point representing it.

Cosine Range- segment from -1 before 1 , since any number of this segment on the x-axis is a projection of any point on the circle, but no point outside this segment is a projection of any of these points.

Cosine period equal to . After all, every time the position of the point representing the number is exactly repeated.

Cosine sign:

1. cosine is equal to zero at , where n- any integer;

2. cosine is positive when , Where n- any integer;

3. cosine is negative when , Where n- any integer.

Cosine- function even. Firstly, the domain of definition of this function is the set of all numbers, and therefore is symmetrical with respect to the origin. And secondly, if we postpone from the beginning two opposite numbers: x And -x, then their abscissas - cosines - will be equal. That is

for anyone x.

1. Cosine increases on segments , Where n- any integer.

2. Cosine decreases on segments , Where n- any integer.

at ;

at .

Tangent

Tangent of a number is called the ratio of the sine of this number to the cosine of this number: .

Tangent angle in A radian is the tangent of a number A.

Tangent- function of number. Her domain- the set of all numbers whose cosine is not equal to zero, since there are no other restrictions in determining the tangent. And since the cosine is equal to zero at , then , Where .

Tangent range

Tangent period x(not equal), differing from each other by , and draw a straight line through them, then this straight line will pass through the origin of coordinates and intersect the line of tangents at some point t. So it turns out that , that is, the number is the period of the tangent.

Tangent sign: tangent is the ratio of sine to cosine. So he

1. is equal to zero when the sine is zero, that is, when , where n- any integer.

2. positive when sine and cosine have identical signs. This happens only in the first and third quarters, that is, when , Where A- any integer.

3. negative when sine and cosine have different signs. This happens only in the second and fourth quarters, that is, when , Where A- any integer.

Tangent- function odd. Firstly, the domain of definition of this function is symmetrical relative to the origin. And secondly, . Due to the oddness of the sine and the evenness of the cosine, the numerator of the resulting fraction is equal to , and its denominator is equal to , which means that this fraction itself is equal to .

So it turned out that .

Means, the tangent increases in each section of its domain of definition, that is, on all intervals of the form , Where A- any integer.

Cotangent

Cotangent of a number is called the ratio of the cosine of this number to the sine of this number: . Cotangent angle in A radian is called the cotangent of a number A. Cotangent- function of number. Her domain- the set of all numbers whose sine is not equal to zero, since there are no other restrictions in the definition of cotangent. And since the sine is equal to zero at , then where

Cotangent range- the set of all real numbers.

Cotangent period equal to . After all, if you take any two valid values x(not equal), differing from each other by , and draw a straight line through them, then this straight line will pass through the origin of coordinates and intersect the line of cotangents at some point t. So it turns out that , that is, that the number is the period of the cotangent.

Counting angles on a trigonometric circle.

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

It is almost the same as in the previous lesson. There are axes, a circle, an angle, everything is in order. Added quarter numbers (in the corners of the large square) - from the first to the fourth. What if someone doesn’t know? As you can see, quarters (they are also called a beautiful word"quadrants") are numbered against the direction clockwise. Added angle values ​​on axes. Everything is clear, no problems.

And a green arrow is added. With a plus. What does it mean? Let me remind you that the fixed side of the angle Always nailed to the positive semi-axis OX. So, if we rotate the movable side of the angle along the arrow with a plus, i.e. in ascending order of quarter numbers, the angle will be considered positive. For example, the picture shows positive angle+60°.

If we put aside the corners V reverse side, clockwise, the angle will be considered negative. Hover your cursor over the picture (or touch the picture on your tablet), you will see a blue arrow with a minus sign. This is the direction of negative angle reading. For example, a negative angle (- 60°) is shown. And you will also see how the numbers on the axes have changed... I also converted them to negative angles. The numbering of the quadrants does not change.

This is where the first misunderstandings usually begin. How so!? What if a negative angle on a circle coincides with a positive one!? And in general, it turns out that the same position of the moving side (or point on the number circle) can be called both a negative angle and a positive one!?

Yes. Exactly. Let's say a positive angle of 90 degrees takes on a circle exactly the same position as a negative angle of minus 270 degrees. A positive angle, for example, +110° degrees takes exactly the same position as negative angle -250°.

No problem. Anything is correct.) The choice of positive or negative angle calculation depends on the conditions of the task. If the condition says nothing in clear text about the sign of the angle, (like "determine the smallest positive angle", etc.), then we work with values ​​that are convenient for us.

An exception (and how could we live without them?!) are trigonometric inequalities, but there we will master this trick.

And now a question for you. How did I know that the position of the 110° angle is the same as the position of the -250° angle?
Let me hint that this is connected with a complete revolution. In 360°... Not clear? Then we draw a circle. We draw it ourselves, on paper. Marking the corner approximately 110°. AND we think, how much time remains until a full revolution. Just 250° will remain...

Got it? And now - attention! If angles 110° and -250° occupy a circle same situation, then what? Yes, the angles are 110° and -250° exactly the same sine, cosine, tangent and cotangent!
Those. sin110° = sin(-250°), ctg110° = ctg(-250°) and so on. Now this is really important! And in itself, there are a lot of tasks where you need to simplify expressions, and as a basis for the subsequent mastery of reduction formulas and other intricacies of trigonometry.

Of course, I took 110° and -250° at random, purely as an example. All these equalities work for any angles occupying the same position on the circle. 60° and -300°, -75° and 285°, and so on. Let me note right away that the angles in these pairs are different. But they have trigonometric functions - the same.

I think you understand what negative angles are. It's quite simple. Counterclockwise - positive counting. Along the way - negative. Consider the angle positive or negative depends on us. From our desire. Well, and also from the task, of course... I hope you understand how to move from negative angles to positive angles and back in trigonometric functions. Draw a circle, an approximate angle, and see how much is missing to complete a full revolution, i.e. up to 360°.

Angles greater than 360°.

Let's deal with angles that are greater than 360°. Are there such things? There are, of course. How to draw them on a circle? No problem! Let's say we need to understand which quarter an angle of 1000° will fall into? Easily! We make one full turn counterclockwise (the angle we were given is positive!). We rewinded 360°. Well, let's move on! One more turn - it’s already 720°. How much is left? 280°. It’s not enough for a full turn... But the angle is more than 270° - and this is the border between the third and fourth quarter. Therefore, our angle of 1000° falls into the fourth quarter. All.

As you can see, it's quite simple. Let me remind you once again that the angle is 1000° and the angle is 280°, which we obtained by discarding the “extra” full revolutions- this, strictly speaking, different corners. But the trigonometric functions of these angles exactly the same! Those. sin1000° = sin280°, cos1000° = cos280°, etc. If I were a sine, I wouldn't notice the difference between these two angles...

Why is all this needed? Why do we need to convert angles from one to another? Yes, all for the same thing.) In order to simplify expressions. Simplification of expressions, in fact, the main task school mathematics. Well, and, along the way, the head is trained.)

Well, let's practice?)

We answer questions. Simple ones first.

1. Which quarter does the -325° angle fall into?

2. Which quarter does the 3000° angle fall into?

3. Which quarter does the angle -3000° fall into?

There is a problem? Or uncertainty? Let's go to Section 555, Practical work with the trigonometric circle. There, in the first lesson of this very " Practical work..." all in detail... In such questions of uncertainty to be shouldn't!

4. What sign does sin555° have?

5. What sign does tg555° have?

Have you determined? Great! Do you have any doubts? You need to go to Section 555... By the way, there you will learn to draw tangent and cotangent on trigonometric circle. A very useful thing.

And now the questions are more sophisticated.

6. Reduce the expression sin777° to the sine of the smallest positive angle.

7. Reduce the expression cos777° to the cosine of the largest negative angle.

8. Reduce the expression cos(-777°) to the cosine of the smallest positive angle.

9. Reduce the expression sin777° to the sine of the largest negative angle.

What, questions 6-9 puzzled you? Get used to it, on the Unified State Exam you don’t find such formulations... So be it, I’ll translate it. Only for you!

The words "bring an expression to..." mean to transform the expression so that its meaning hasn't changed A appearance changed according to the assignment. So, in tasks 6 and 9 we must get a sine, inside of which there is smallest positive angle. Everything else doesn't matter.

I will give out the answers in order (in violation of our rules). But what to do, there are only two signs, and there are only four quarters... You won’t be spoiled for choice.

6. sin57°.

7. cos(-57°).

8. cos57°.

9. -sin(-57°)

I assume that the answers to questions 6-9 confused some people. Especially -sin(-57°), really?) Indeed, in elementary rules when calculating angles there is room for errors... That is why I had to do a lesson: “How to determine the signs of functions and bring angles on a trigonometric circle?” In Section 555. Tasks 4 - 9 are covered there. Well sorted, with all the pitfalls. And they are here.)

In the next lesson we will deal with the mysterious radians and the number "Pi". Let's learn how to easily and correctly convert degrees to radians and vice versa. And we will be surprised to discover that this basic information on the website enough already to solve some custom trigonometry problems!

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