If you are already familiar with trigonometric circle , and you just want to refresh the memory of 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 associate with impassable thickets. Suddenly, there are so many values \u200b\u200bof trigonometric functions, so many formulas ... But it was, like, it went wrong at first, and ... off we go ... complete misunderstanding ...
It is very important not to wave your hand at values \u200b\u200bof trigonometric functions, - they say, you can always look in the spur with a table of values.
If you constantly look at the table with the values \u200b\u200bof trigonometric formulas, let's get rid of this habit!
Will help us out! You will work with it several times, and then it will pop up in your head. Why is it better than a table? Yes, in the table you will find a limited number of values, but on the circle - EVERYTHING!
For example, tell by looking in standard table of trigonometric formula values what equal to sinesay 300 degrees, or -45.
No way? .. you can, of course, connect reduction formulas… And looking at the trigonometric circle, one can easily answer such questions. And you will soon know how!
And when deciding trigonometric equations and inequalities without a trigonometric circle - nowhere at all.
Introducing the trigonometric circle
Let's go in order.
First, let's write out the following series of numbers:
And now this:
And finally, like this:
Of course, it is clear that, in fact, it is in the first place, in the second place, and in the last -. That is, we will be more interested in the chain.
But how beautiful it turned out! In which case, we will restore this "miraculous ladder".
And why do we need it?
This chain is the main values \u200b\u200bof sine and cosine in the first quarter.
Let's draw a circle of unit radius in a rectangular coordinate system (that is, we take any radius along the length, and declare its length to be unit).
From the "0-Start" ray, set aside the angles in the direction of the arrow (see Fig.).
We get the corresponding points on the circle. So if we project the points on each of the axes, then we will come out just at the values \u200b\u200bfrom the above chain.
Why is that, you ask?
We will not analyze everything. Consider principlethat will allow you to cope with other, similar situations.
Triangle AOB - rectangular, in it. And we know that opposite the angle b lies a leg half the size of the hypotenuse (we have the hypotenuse \u003d the radius of the circle, that is, 1).
Hence, AB \u003d (and hence OM \u003d). And according to the Pythagorean theorem
I hope that something is already becoming clear?
So point B will correspond to the value, and point M - to the value
Likewise with the rest of the values \u200b\u200bof the first quarter.
As you understand, the axis (ox) familiar to us will be the cosine axis, and the (oy) axis is sine axis ... later.
To the left of zero on the cosine axis (below zero on the sine axis) there will of course be negative values.
So, here he is, the Omnipotent, without which there is nowhere in trigonometry.
But how to use the trigonometric circle, we will talk in.
Back forward
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Purpose: teach how to use the unit circle when solving various trigonometric tasks.
In the school mathematics course, various options for the introduction of trigonometric functions are possible. The most convenient and commonly used is the "numeric unit circle". Its application in the topic "Trigonometry" is very extensive.
The unit circle is used for:
- determination of sine, cosine, tangent and cotangent of an angle;
- finding the values \u200b\u200bof trigonometric functions for some values \u200b\u200bof the numerical and angular argument;
- derivation of the basic formulas of trigonometry;
- derivation of reduction formulas;
- finding the domain of definition and the range of values \u200b\u200bof trigonometric functions;
- determination of the periodicity of trigonometric functions;
- determination of parity and oddness of trigonometric functions;
- determination of intervals of increase and decrease of trigonometric functions;
- determination of intervals of constancy of trigonometric functions;
- radian angle measurement;
- finding the values \u200b\u200bof inverse trigonometric functions;
- solution of the simplest trigonometric equations;
- solution of the simplest inequalities, etc.
Thus, active, conscious mastery of this type of visualization by students provides undeniable advantages for mastering the section of mathematics "Trigonometry".
The use of ICT in the lessons of teaching mathematics makes it easier to master the numerical unit circle. Of course, interactive whiteboard has a wide range of applications, but not all classes have it. If we talk about the use of presentations, then on the Internet and their choice is great, and each teacher can find the most appropriate option for their lessons.
What is special about this presentation?
This presentation assumes various use cases and is not illustrative for a specific lesson in the topic "Trigonometry". Each slide of this presentation can be used in isolation, both at the stage of explaining the material, developing skills, and for reflection. When creating this presentation, special attention was paid to its "readability" from a distance, since the number of students with reduced vision is constantly growing. A well-thought-out color scheme, logically connected objects are united by a single color. The presentation is animated in such a way that the teacher can comment on a fragment of the slide, and the student can ask a question. Thus, this presentation is a kind of "movable" tables. The last slides are not animated and are used to check the assimilation of the material in the course of solving trigonometric tasks. The circle on the slides is simplified as much as possible outwardly and is as close as possible to the one shown on the notebook sheet by the students. I consider this condition to be fundamental. It is important for students to form an opinion about the unit circle as an accessible and mobile (although not the only) form of visibility when solving trigonometric tasks.
This presentation will help teachers familiarize students with the unit circle in grade 9 in geometry lessons when studying the topic "Relationships between sides and angles of a triangle." And, of course, it will help to expand and deepen the skill of working with the unit circle when solving trigonometric problems for senior students in algebra lessons.
Slides 3, 4explain the construction of the unit circle; the principle of determining the location of a point on the unit circle in I and II coordinate quarters; transition from geometric definitions of sine and cosine functions (in a right-angled triangle) to algebraic ones on the unit circle.
Slides 5-8 explain how to find the values \u200b\u200bof trigonometric functions for the main angles of the I coordinate quarter.
Slides 9-11explains the signs of functions in coordinate quarters; determination of intervals of constancy of trigonometric functions.
Slide 12 used to form ideas about positive and negative angles; acquaintance with the concept of periodicity of trigonometric functions.
Slides 13, 14 used when converting to a radian measure of an angle.
Slides 15-18 not animated and are used when solving various trigonometric tasks, consolidating and checking the results of mastering the material.
- Title page.
- Goal setting.
- Creates a unit circle. The main values \u200b\u200bof the angles in degrees.
- Determination of the sine and cosine of an angle on the unit circle.
- Table values \u200b\u200bfor sine in ascending order.
- The table values \u200b\u200bfor cosine are in ascending order.
- The table values \u200b\u200bfor the tangent are in ascending order.
- Table values \u200b\u200bfor cotangent in ascending order.
- Function signs sin α.
- Function signs cos α.
- Function signs tg αand ctg α.
- Positive and negative angles on the unit circle.
- Radian measure of angle.
- Positive and negative angles in radians on the unit circle.
- Various variants of the unit circle to consolidate and check the results of mastering the material.
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Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and star orientation. These calculations were related to spherical trigonometry, while in the school course they study the aspect ratio and angle of a flat triangle.
Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationship between the sides and angles of triangles.
During the heyday of culture and science of 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-Marazvi introduced functions such as tangent and cotangent, compiled the first tables of values \u200b\u200bfor sines, tangents and cotangents. The concept of sine and cosine was introduced by Indian scientists. A lot of attention is devoted to trigonometry 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 \u200b\u200bof these quantities are based on the Pythagorean theorem. Schoolchildren know it better in the wording: "Pythagorean pants, equal in all directions", since the proof is given on the example of an isosceles right-angled triangle.
Sine, cosine and other dependencies establish a relationship between acute angles and the sides of any right triangle. Let's give formulas for calculating these values \u200b\u200bfor angle A and trace the relationship of trigonometric functions:
As you can see, tg and ctg are inverse functions... If we represent leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, then we get the following formulas for tangent and cotangent:
Trigonometric circle
Graphically, the ratio of these quantities can be represented as follows:
The circle, in this case, represents all possible values \u200b\u200bof the angle α - from 0 ° to 360 °. As you can see from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will be with a "+" sign if α belongs to the I and II quarters of the circle, that is, is in the range from 0 ° to 180 °. When α is from 180 ° to 360 ° (III and IV quarters), sin α can only be negative.
Let's try to build trigonometric tables for specific angles and find out the value of the quantities.
The values \u200b\u200bof α equal to 30 °, 45 °, 60 °, 90 °, 180 ° and so on are called special cases. The values \u200b\u200bof trigonometric functions for them are calculated and presented in the form of special tables.
These angles are not chosen by chance. The designation π in the tables stands for radians. Rad is the angle at which the length of a circular 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.
The angles in the tables for trigonometric functions correspond to the values \u200b\u200bof radians:
So, it's not hard to guess that 2π is a full circle or 360 °.
Properties of trigonometric functions: sine and cosine
In order to consider and compare the main 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 a comparative table of properties for a sine wave and a cosine wave:
| Sinusoid | Cosine |
|---|---|
| y \u003d sin x | y \u003d cos x |
| ODZ [-1; one] | ODZ [-1; one] |
| sin x \u003d 0, for x \u003d πk, where k ϵ Z | cos x \u003d 0, for x \u003d π / 2 + πk, where k ϵ Z |
| sin x \u003d 1, for x \u003d π / 2 + 2πk, where k ϵ Z | cos x \u003d 1, for x \u003d 2πk, where k ϵ Z |
| sin x \u003d - 1, for x \u003d 3π / 2 + 2πk, where k ϵ Z | cos x \u003d - 1, for x \u003d π + 2πk, where k ϵ Z |
| sin (-x) \u003d - sin x, that is, the function is odd | cos (-x) \u003d cos x, i.e. the function is even |
| the function is periodic, the smallest period is 2π | |
| sin x ›0, for x belonging to I and II quarters or from 0 ° to 180 ° (2πk, π + 2πk) | cos x ›0, for x belonging to I and IV quarters or from 270 ° to 90 ° (- π / 2 + 2πk, π / 2 + 2πk) |
| sin x ‹0, for x belonging to the III and IV quarters or from 180 ° to 360 ° (π + 2πk, 2π + 2πk) | cos x ‹0, with x belonging to the II and III quarters or from 90 ° to 270 ° (π / 2 + 2πk, 3π / 2 + 2πk) |
| increases on the interval [- π / 2 + 2πk, π / 2 + 2πk] | increases on the interval [-π + 2πk, 2πk] |
| decreases on the intervals [π / 2 + 2πk, 3π / 2 + 2πk] | decreases in intervals |
| derivative (sin x) ’\u003d cos x | derivative (cos x) ’\u003d - sin x |
Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with signs of trigonometric quantities and mentally "add up" the graph about the OX axis. If the signs coincide, the function is even, otherwise it is odd.
The introduction of radians and the enumeration of the main properties of a sinusoid and cosine allow us to give the following pattern:
It is very easy to make sure that the formula is correct. For example, for x \u003d π / 2 the sine is 1, as is the cosine x \u003d 0. The check can be carried out by referring to tables or by tracing the curves of functions for given values.
Tangentoid and Cotangentoid Properties
Plots of tangent and cotangent functions differ significantly from sine and cosine. The tg and ctg values \u200b\u200bare inverse to each other.
- Y \u003d tg x.
- The tangensoid tends to the y-values \u200b\u200bat x \u003d π / 2 + πk, but never reaches them.
- The smallest positive period of the tangentoid is π.
- Tg (- x) \u003d - tg x, that is, the function is odd.
- Tg x \u003d 0, for x \u003d 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) ’\u003d 1 / cos 2 \u2061x.
Consider a graphical representation of a cotangentoid below in the text.
The main properties of a cotangensoid:
- Y \u003d ctg x.
- Unlike sine and cosine functions, in the tangentoid Y can take on the values \u200b\u200bof the set of all real numbers.
- The cotangensoid tends to the values \u200b\u200bof y at x \u003d πk, but never reaches them.
- The smallest positive period of a cotangensoid is π.
- Ctg (- x) \u003d - ctg x, that is, the function is odd.
- Ctg x \u003d 0, for x \u003d 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) ’\u003d - 1 / sin 2 \u2061x Correct