Examples:

\(\sin(⁡30^°)=\)\(\frac(1)(2)\)
\(\sin⁡\)\(\frac(π)(3)\) \(=\)\(\frac(\sqrt(3))(2)\)
\(\sin⁡2=0.909…\)

Argument and meaning

Sine of an acute angle

Sine of an acute angle can be determined using a right triangle - it is equal to the ratio of the opposite side to the hypotenuse.

Example :

1) Let an angle be given and you need to determine the sine of this angle.

2) Let us complete any right triangle on this angle.

3) Having measured the required sides, we can calculate \(sinA\).

Sine of a number

The number circle allows you to determine the sine of any number, but usually you find the sine of numbers somehow related to: \(\frac(π)(2)\) , \(\frac(3π)(4)\) , \(-2π\ ).

For example, for the number \(\frac(π)(6)\) - the sine will be equal to \(0.5\). And for the number \(-\)\(\frac(3π)(4)\) it will be equal to \(-\)\(\frac(\sqrt(2))(2)\) (approximately \(-0 ,71\)).

For sine for other numbers often encountered in practice, see.

The sine value always lies in the range from \(-1\) to \(1\). Moreover, it can be calculated for absolutely any angle and number.

Sine of any angle

Thanks to the unit circle, it is possible to determine trigonometric functions not only of an acute angle, but also of 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: let us need to define \(sin∠KOA\) with the 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 \(\sin⁡∠KOA\).

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 sine of a number and the sine of an arbitrary angle are defined almost identically. Only the way the point is found on the circle changes.

Relation to other trigonometric functions:

Function \(y=\sin⁡x\)

If we plot the angles in radians along the \(x\) axis, and the sine values ​​corresponding to these angles along the \(y\) axis, we get the following graph:

This graph is called a sine wave and has the following properties:

The domain of definition is any value of x: \(D(\sin⁡x)=R\)
- range of values ​​– from \(-1\) to \(1\) inclusive: \(E(\sin⁡x)=[-1;1]\)
- odd: \(\sin⁡(-x)=-\sin⁡x\)
- periodic with period \(2π\): \(\sin⁡(x+2π)=\sin⁡x\)
- points of intersection with coordinate axes:
abscissa axis: \((πn;0)\), where \(n ϵ Z\)
Y axis: \((0;0)\)
- intervals of constancy of sign:
the function is positive on the intervals: \((2πn;π+2πn)\), where \(n ϵ Z\)
the function is negative on the intervals: \((π+2πn;2π+2πn)\), where \(n ϵ Z\)
- intervals of increase and decrease:
the function increases on the intervals: \((-\)\(\frac(π)(2)\) \(+2πn;\) \(\frac(π)(2)\) \(+2πn)\), where \(n ϵ Z\)
the function decreases on the intervals: \((\)\(\frac(π)(2)\) \(+2πn;\)\(\frac(3π)(2)\) \(+2πn)\), where \(n ϵ Z\)
- maximums and minimums of the function:
the function has a maximum value \(y=1\) at points \(x=\)\(\frac(π)(2)\) \(+2πn\), where \(n ϵ Z\)
the function has a minimum value \(y=-1\) at points \(x=-\)\(\frac(π)(2)\) \(+2πn\), where \(n ϵ Z\).

One of the most frequently used of all Bradis trigonometric tables is the table of sines. In this article we will understand the concept of sine (sin), learn how to find sine values ​​for various angles (0, 30, 45, 60, 90), and understand why a table of sines is needed.

Table of sines and its application

First, we need to remind you what the concept of sine of an angle means.

Sine - this is the ratio of the leg opposite this angle to the hypotenuse.

This is true if the triangle is right-angled.

Standard right triangle: sides a (BC) and b (AC) are legs, side c (AB) is the hypotenuse

Example: find the sine of angle ⍺ ​​and angle β

sin ⍺ = a/c or the ratio of side BC to side AB. If we take angle β, then side b or AC will be considered opposite. The hypotenuse in this case is the same - AB. Then:

sin β = b/s or AC relation AB.

In a right triangle always 2 legs but only one hypotenuse

As you know, there are 360 ​​integer angle values. But often you need to calculate the values ​​for the most popular angles, such as: sine 0°, sine 30°, sine 45°, sine 60°, sine 90°. These values ​​can be found in the Bradis tables.

Despite the fact that in 2021 it celebrates its centenary, the Bradis table has not lost its relevance. In particular, it is used by architects, designers, and constructors to carry out quick intermediate calculations. Bradis tables are approved for use in schools when taking the Unified State Exam, unlike calculators.

Online calculator for calculating the sine of an angle

Table of values ​​of trigonometric functions

Note. This table of trigonometric function values ​​uses the √ sign to represent the square root. To indicate a fraction, use the symbol "/".

see also useful materials:

For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, sine 30 degrees - we look for the column with the heading sin (sine) and find the intersection of this table column with the row “30 degrees”, at their intersection we read the result - one half. Similarly we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin column and the 60 degree line we find the value sin 60 = √3/2), etc. The values ​​of sines, cosines and tangents of other “popular” angles are found in the same way.

Sine pi, cosine pi, tangent pi and other angles in radians

The table below of cosines, sines and tangents is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the angle of 60 degrees in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi unambiguously expresses the dependence of the circumference on the degree measure of the angle. Thus, pi radians are equal to 180 degrees.

Any number expressed in terms of pi (radians) can be easily converted to degrees by replacing pi (π) with 180.

Examples:
1. Sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and it is equal to zero.

2. Cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and it is equal to minus one.

3. Tangent pi
tg π = tg 180 = 0
thus, tangent pi is the same as tangent 180 degrees and it is equal to zero.

Table of sine, cosine, tangent values ​​for angles 0 - 360 degrees (common values)

If in the table of values ​​of trigonometric functions a dash is indicated instead of the function value (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle the function does not have a specific value. If there is no dash, the cell is empty, which means we have not yet entered the required value. We are interested in what queries users come to us for and supplement the table with new values, despite the fact that current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is quite sufficient to solve most problems.

Table of values ​​of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numeric values ​​“as per Bradis tables”)

As you can see, this circle is constructed in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin of coordinates, the initial position of 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 revolution and stop at position or.

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 for finding 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, the 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, the desired point will be in the same position as when turning to. 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 a negative value and the sine takes a positive value, we have:

Such examples are discussed in more detail when studying the 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.