Cosine 90 degrees. What is the cosine of an angle and how to use it in solving problems

In the article, we will fully understand what it looks like table of trigonometric values, sine, cosine, tangent and cotangent. Let's consider the basic meaning of trigonometric functions, from an angle of 0,30,45,60,90,...,360 degrees. And let's see how to use these tables in calculating the values ​​of trigonometric functions.
First let's look at table of cosine, sine, tangent and cotangent from an angle of 0, 30, 45, 60, 90,... degrees. The definition of these quantities allows us to determine the value of the functions of angles of 0 and 90 degrees:

sin 0 0 =0, cos 0 0 = 1. tg 0 0 = 0, cotangent from 0 0 will be undefined
sin 90 0 = 1, cos 90 0 =0, ctg90 0 = 0, tangent from 90 0 will be uncertain

If you take right triangles whose angles are from 30 to 90 degrees. We get:

sin 30 0 = 1/2, cos 30 0 = √3/2, tan 30 0 = √3/3, cos 30 0 = √3
sin 45 0 = √2/2, cos 45 0 = √2/2, tan 45 0 = 1, cos 45 0 = 1
sin 60 0 = √3/2, cos 60 0 = 1/2, tg 60 0 =√3, cot 60 0 = √3/3

Let us represent all the obtained values ​​in the form trigonometric table:

Table of sines, cosines, tangents and cotangents!

If we use the reduction formula, our table will increase, adding values ​​for angles up to 360 degrees. It will look like:

Also, based on the properties of periodicity, the table can be increased if we replace the angles with 0 0 +360 0 *z .... 330 0 +360 0 *z, in which z is an integer. In this table it is possible to calculate the value of all angles corresponding to points in a single circle.

Let's look at how to use the table in a solution.
Everything is very simple. Since the value we need lies at the intersection point of the cells we need. For example, take the cos of an angle of 60 degrees, in the table it will look like:

In the final table of the main values ​​of trigonometric functions, we proceed in the same way. But in this table it is possible to find out how much the tangent from an angle of 1020 degrees is, it = -√3 Let's check 1020 0 = 300 0 +360 0 *2. Let's find it using the table.

For more searching, trigonometric angle values ​​accurate to minutes are used. Detailed instructions on how to use them are on the page.

Bradis table. For sine, cosine, tangent and cotangent.

The Bradis tables are divided into several parts, consisting of tables of cosine and sine, tangent and cotangent - which is divided into two parts (tg of angles up to 90 degrees and ctg of small angles).

Sine and cosine

tg of the angle starting from 0 0 ending with 76 0, ctg of the angle starting from 14 0 ending with 90 0.

tg up to 90 0 and ctg of small angles.

Let's figure out how to use Bradis tables in solving problems.

Let's find the designation sin (designation in the column on the left edge) 42 minutes (designation is on the top line). By intersection we look for the designation, it = 0.3040.

The minute values ​​are indicated with an interval of six minutes, what to do if the value we need falls exactly within this interval. Let's take 44 minutes, but there are only 42 in the table. We take 42 as a basis and use the additional columns on the right side, take the 2nd amendment and add to 0.3040 + 0.0006 we get 0.3046.

With sin 47 minutes, we take 48 minutes as a basis and subtract 1 correction from it, i.e. 0.3057 - 0.0003 = 0.3054

When calculating cos, we work similarly to sin, only we take the bottom row of the table as a basis. For example cos 20 0 = 0.9397

The values ​​of tg angle up to 90 0 and cot of a small angle are correct and there are no corrections in them. For example, find tg 78 0 37min = 4.967


and ctg 20 0 13min = 25.83

Well, we've looked at the basic trigonometric tables. We hope this information was extremely useful to you. If you have any questions about the tables, be sure to write them in the comments!

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Tables of values ​​of sines (sin), cosines (cos), tangents (tg), cotangents (ctg) are a powerful and useful tool that helps solve many problems, both theoretical and applied. In this article we will provide a table of basic trigonometric functions (sines, cosines, tangents and cotangents) for angles of 0, 30, 45, 60, 90, ..., 360 degrees (0, π 6, π 3, π 2,... . , 2 π radians). Separate Bradis tables for sines and cosines, tangents, and cotangents will also be shown, with an explanation of how to use them to find the values ​​of basic trigonometric functions.

Table of basic trigonometric functions for angles 0, 30, 45, 60, 90, ..., 360 degrees

Based on the definitions of sine, cosine, tangent and cotangent, you can find the values ​​of these functions for angles of 0 and 90 degrees

sin 0 = 0, cos 0 = 1, t g 0 = 0, zero cotangent is not defined,

sin 90° = 1, cos 90° = 0, c t g 90° = 0, tangent of ninety degrees is not defined.

The values ​​of sines, cosines, tangents and cotangents in the geometry course are defined as the ratio of the sides of a right triangle, the angles of which are 30, 60 and 90 degrees, and also 45, 45 and 90 degrees.

Defining trigonometric functions for an acute angle in a right triangle

Sinus- the ratio of the opposite side to the hypotenuse.

Cosine- the ratio of the adjacent leg to the hypotenuse.

Tangent- the ratio of the opposite side to the adjacent side.

Cotangent- the ratio of the adjacent side to the opposite side.

In accordance with the definitions, the values ​​of the functions are found:

sin 30 ° = 1 2 , cos 30 ° = 3 2 , t g 30 ° = 3 3 , c t g 30 ° = 3 , sin 45 ° = 2 2 , cos 45 ° = 2 2 , t g 45 ° = 1 , c t g 45 ° = 1, sin 60° = 3 2, cos 45° = 1 2, tg 45° = 3, c tg 45° = 3 3.

Let's put these values ​​in a table and call it a table of the basic values ​​of sine, cosine, tangent and cotangent.

Table of basic values ​​of sines, cosines, tangents and cotangents

One of the important properties of trigonometric functions is periodicity. Based on this property, this table can be expanded using reduction formulas. Below we present an extended table of the values ​​of the main trigonometric functions for angles 0, 30, 60, ... , 120, 135, 150, 180, ... , 360 degrees (0, π 6, π 3, π 2, ... , 2 π radians).

Table of sines, cosines, tangents and cotangents

The periodicity of sine, cosine, tangent and cotangent allows you to expand this table to arbitrarily large angle values. The values ​​collected in the table are used most often when solving problems, so it is recommended to memorize them.

How to use the table of basic values ​​of trigonometric functions

The principle of using a table of values ​​of sines, cosines, tangents and cotangents is clear on an intuitive level. The intersection of a row and a column gives the value of the function for a particular angle.

Example. How to use the table of sines, cosines, tangents and cotangents

We need to find out what sin 7 π 6 is equal to

We find a column in the table whose last cell value is 7 π 6 radians - the same as 210 degrees. Then we select the term of the table in which the values ​​of sines are presented. At the intersection of the row and column we find the desired value:

sin 7 π 6 = - 1 2

Bradis tables

The Bradis table allows you to calculate the value of sine, cosine, tangent or cotangent with an accuracy of 4 decimal places without the use of computer technology. This is a kind of replacement for an engineering calculator.

Reference

Vladimir Modestovich Bradis (1890 - 1975) - Soviet mathematician-teacher, since 1954 corresponding member of the Academy of Pedagogical Sciences of the USSR. Tables of four-digit logarithms and natural trigonometric quantities developed by Bradis were first published in 1921.

First, we present the Bradis table for sines and cosines. It allows you to quite accurately calculate the approximate values ​​of these functions for angles containing an integer number of degrees and minutes. The leftmost column of the table represents degrees, and the top row represents minutes. Note that all angle values ​​of the Bradis table are multiples of six minutes.

Bradis table for sines and cosines

To find the values ​​of sines and cosines of angles not presented in the table, it is necessary to use corrections.

Now we present the Bradis table for tangents and cotangents. It contains values ​​of tangents of angles from 0 to 76 degrees, and cotangents of angles from 14 to 90 degrees.

Bradis table for tangent and cotangent

How to use Bradis tables

Consider the Bradis table for sines and cosines. Everything related to sinuses is at the top and to the left. If we need cosines, look at the right side at the bottom of the table.

To find the values ​​of the sine of an angle, you need to find the intersection of the row containing the required number of degrees in the leftmost cell and the column containing the required number of minutes in the top cell.

If the exact angle value is not in the Bradis table, we resort to corrections. Corrections for one, two and three minutes are given in the rightmost columns of the table. To find the value of the sine of an angle that is not in the table, we find the value closest to it. After this, we add or subtract the correction corresponding to the difference between the angles.

If we are looking for the sine of an angle that is greater than 90 degrees, we first need to use the reduction formulas, and only then the Bradis table.

Example. How to use the Bradis table

Let's say we need to find the sine of the angle 17 ° 44 ". Using the table, we find what the sine of 17 ° 42 " is equal to and add a correction of two minutes to its value:

17°44" - 17°42" = 2" (necessary correction) sin 17°44" = 0. 3040 + 0 . 0006 = 0 . 3046

The principle of working with cosines, tangents and cotangents is similar. However, it is important to remember the sign of the amendments.

Important!

When calculating the values ​​of sines, the correction has a positive sign, and when calculating cosines, the correction must be taken with a negative sign.

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Cosine table is a convenient solution for making quick calculations when you need to get the numerical value of the cosine of a particular angle. In the article we will learn what a cosine is, how the tables of sines and cosines are similar and how they are related, how to use the Bradis table of sines to obtain specific numerical values ​​of the cosine of a particular angle.

What is the cosine of an angle and how to use it in solving problems

Let's start with the fact that everyone knows what a right triangle is. It is called a triangle in which one of the angles (C) is straight (equal to 90°), the other two angles (? and?) are acute. It has a standard designation for angles and sides. Then what is the cosine of an angle can be considered further.

Right triangle: sides a (BC) and b (AC) - legs, side c (AB) - hypotenuse

A right angle is always equal to 90°, an acute angle is always less, and an obtuse angle is always more than 90°.

Cosine — This is the ratio of the adjacent side to the hypotenuse:

• cos α = b divided by c;
• cos β = а(BC)/с(AB).

That is, if you need to know, for example, what height to make the roof above the house, if you know the width of the house and the angle of inclination of the roof so that the snow does not linger, then it will not be difficult to calculate the height of the ridge using the cosine theorem. It must be remembered that functions such as cosines and sines in formulas depend on the angle. Sine works on the opposite side, cosine works on the adjacent side.

These are trigonometric formulas for calculating angles in a triangle using trigonometric functions, cosine, tangent, cotangent

Cosine - the ratio of the adjacent leg to the hypotenuse

If the triangle is not right-angled, its parameters can also be calculated using Euclid's theorem. Its essence is that a triangle lying on a plane and having sides a, b, c, as well as an angle α, which is opposite side a, can be calculated using the following formula:

а²= b²+с²-2²· b· cos α or:

From here we can find cos α, cos α = (b²+2²- a²) : 2bс.

A small clarification: if the angle α is less than 90°, then b²+2²- a² > 0, if α =90°, then b²+2²- a²=0, if α >90°, that is, the angle is obtuse, then b²+2² - a²< 0.

We do the same calculations for other angles of the triangle:

• c² = a² + b² - 2ab cosγ,
• b² = a² + c² - 2ac cosβ.

How to calculate the cosine of an angle without formulas

There are some angles whose cosine can be calculated without formulas, using table of sines and cosines π . In it, the calculation is carried out through the number π, which is divided by an integer, depending on the size of the angle, that is, sin 30° = π: 6 or 0.5, cos 30° = √3: 2. This table contains cosine data 30 degrees, cosine 45 degrees, cosine 60 degrees, cosine 90 degrees, cosine 120 degrees, cosine 180 degrees, cosine 270 degrees, cosine 360 ​​degrees, cosine 0, as well as similar values ​​of sines.

Below is a table of cosines, additionally the sines are indicated in their numerical expression.

Online cosine calculator