Coordinates x points lying on the circle are equal to cos(θ), and the coordinates y correspond to sin(θ), where θ is the magnitude of the angle.
- If you find it difficult to remember this rule, just remember that in the pair (cos; sin) “the sine comes last.”
- This rule can be derived by considering right triangles and the definition of these trigonometric functions (the sine of an angle is equal to the ratio of the length of the opposite side, and the cosine of the adjacent side to the hypotenuse).
Write down the coordinates of four points on the circle. A “unit circle” is a circle whose radius is equal to one. Use this to determine the coordinates x And y at four points of intersection of the coordinate axes with the circle. Above, for clarity, we designated these points as “east”, “north”, “west” and “south”, although they do not have established names.
- "East" corresponds to the point with coordinates (1; 0) .
- "North" corresponds to the point with coordinates (0; 1) .
- "West" corresponds to the point with coordinates (-1; 0) .
- "South" corresponds to the point with coordinates (0; -1) .
- This is similar to a regular graph, so there is no need to memorize these values, just remember the basic principle.
Remember the coordinates of the points in the first quadrant. The first quadrant is located in the upper right part of the circle, where the coordinates x And y take positive values. These are the only coordinates you need to remember:
Draw straight lines and determine the coordinates of the points of their intersection with the circle. If you draw straight horizontal and vertical lines from the points of one quadrant, the second points of intersection of these lines with the circle will have the coordinates x And y with the same absolute values, but different signs. In other words, you can draw horizontal and vertical lines from the points of the first quadrant and label the points of intersection with the circle with the same coordinates, but at the same time leave space on the left for the correct sign ("+" or "-").
To determine the sign of the coordinates, use the rules of symmetry. There are several ways to determine where to place the "-" sign:
- Remember the basic rules for regular charts. Axis x negative on the left and positive on the right. Axis y negative from below and positive from above;
- start with the first quadrant and draw lines to other points. If the line crosses the axis y, coordinate x will change its sign. If the line crosses the axis x, the sign of the coordinate will change y;
- remember that in the first quadrant all functions are positive, in the second quadrant only the sine is positive, in the third quadrant only the tangent is positive, and in the fourth quadrant only the cosine is positive;
- Whichever method you use, you should get (+,+) in the first quadrant, (-,+) in the second, (-,-) in the third, and (+,-) in the fourth.
Check if you made a mistake. Below is a complete list of coordinates of “special” points (except for the four points on the coordinate axes), if you move along the unit circle counterclockwise. Remember that to determine all these values, it is enough to remember the coordinates of the points only in the first quadrant:
- first quadrant: ( 3 2 , 1 2 (\displaystyle (\frac (\sqrt (3))(2)),(\frac (1)(2)))); (2 2 , 2 2 (\displaystyle (\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (1 2 , 3 2 (\displaystyle (\frac (1)(2)),(\frac (\sqrt (3))(2))));
- second quadrant: ( − 1 2 , 3 2 (\displaystyle -(\frac (1)(2)),(\frac (\sqrt (3))(2)))); (− 2 2 , 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),(\frac (\sqrt (2))(2)))); (− 3 2 , 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),(\frac (1)(2))));
- third quadrant: ( − 3 2 , − 1 2 (\displaystyle -(\frac (\sqrt (3))(2)),-(\frac (1)(2)))); (− 2 2 , − 2 2 (\displaystyle -(\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (− 1 2 , − 3 2 (\displaystyle -(\frac (1)(2)),-(\frac (\sqrt (3))(2))));
- fourth quadrant: ( 1 2 , − 3 2 (\displaystyle (\frac (1)(2)),-(\frac (\sqrt (3))(2)))); (2 2 , − 2 2 (\displaystyle (\frac (\sqrt (2))(2)),-(\frac (\sqrt (2))(2)))); (3 2 , − 1 2 (\displaystyle (\frac (\sqrt (3))(2)),-(\frac (1)(2)))).
If you are already familiar with trigonometric circle , and you just want to refresh your memory of certain 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, so many values of trigonometric functions, so many formulas pile up... But it’s like, it didn’t work out at the beginning, and... off we go... 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 are constantly looking at a table with the values of 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. How 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, say while looking at standard table of values of trigonometric formulas , what is the sine equal to, 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 solving trigonometric equations and inequalities without a trigonometric circle, it’s absolutely nowhere.
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 one:
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 us draw a circle of unit radius in a rectangular coordinate system (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. 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, 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
Trigonometric circle. Unit circle. Number circle. What it is?
Attention!
There are additional
materials in Special section 555.
For those who are very "not very..."
And for those who “very much…”)
Very often terms trigonometric circle, unit circle, number circle poorly understood by students. And completely in vain. These concepts are a powerful and universal assistant in all areas of trigonometry. In fact, this is a legal cheat sheet! I drew a trigonometric circle and immediately saw the answers! Tempting? So let's learn, it would be a sin not to use such a thing. Moreover, it is not at all difficult.
To successfully work with the trigonometric circle, you need to know only three things.
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