In this article we will establish the relationship between the basic units of measurement of angles - degrees and radians. This connection will ultimately allow us to carry out converting degrees to radians and back. So that these processes do not cause difficulties, we will obtain a formula for converting degrees to radians and a formula for converting from radians to degrees, after which we will analyze in detail the solutions to the examples.

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Relationship between degrees and radians

The connection between degrees and radians will be established if both the degree and radian measures of an angle are known (the degree and radian measures of an angle can be found in the section).

Let us take the central angle based on the diameter of a circle of radius r. We can calculate the measure of this angle in radians: to do this we need to divide the length of the arc by the length of the radius of the circle. This angle corresponds to an arc length equal to half circumference, that is, . Dividing this length by the length of the radius r, we obtain the radian measure of the angle we took. So our angle is rad. On the other hand, this angle is expanded, it is equal to 180 degrees. Therefore, pi radians is 180 degrees.

So, it is expressed by the formula π radians = 180 degrees, that is, .

Formulas for converting degrees to radians and radians to degrees

From the equality of the form , which we obtained in the previous paragraph, we can easily deduce formulas for converting radians to degrees and degrees to radians.

Dividing both sides of the equality by pi, we obtain a formula expressing one radian in degrees: . This formula means that the degree measure of an angle of one radian is equal to 180/π. If we swap the left and right sides of the equality and then divide both sides by 180, we get a formula of the form . It expresses one degree in radians.

To satisfy our curiosity, let's calculate the approximate value of an angle of one radian in degrees and the value of an angle of one degree in radians. To do this, take the value of pi accurate to ten thousandths and substitute it into the formulas And , and carry out the calculations. We have And . So, one radian is approximately equal to 57 degrees, and one degree is 0.0175 radians.

Finally, from the obtained relations And Let's move on to the formulas for converting radians to degrees and vice versa, and also consider examples of the application of these formulas.

Formula for converting radians to degrees has the form: . Thus, if the value of the angle in radians is known, then multiplying it by 180 and dividing by pi, we obtain the value of this angle in degrees.

Example.

An angle of 3.2 radians is given. What is the measure of this angle in degrees?

Solution.

Let's use the formula for converting from radians to degrees, we have

Answer:

.

Formula for converting degrees to radians looks like . That is, if the value of the angle in degrees is known, then multiplying it by pi and dividing by 180, we obtain the value of this angle in radians. Let's look at the example solution.

The need to measure angles has appeared among people since civilization reached a minimum technical level. Everyone knows the phenomenal accuracy of observance of the inclination and orientation according to the cardinal points provided by the builders Egyptian pyramids. The modern degree measure of angles is now believed to have been invented by the ancient Akkadians.

What are degrees?

A degree is a generally accepted unit of measurement for angles. In a complete circle 360 ​​degrees. The reason for choosing this particular number is unknown. The Akkadians probably divided the circle into sectors using the angle of an equilateral triangle, and then divided the resulting segments again into 60 parts according to their number system. A degree is also divided into 60 minutes, and minutes into 60 seconds. Commonly accepted designations are:

° - angular degrees

’ - minutes,

'' - seconds.

Over the millennia, the degree measure of angles has become firmly established in many areas of human activity. It is still indispensable in all areas of science and technology - from cartography to calculating the orbits of artificial Earth satellites.

What are radians?

Archimedes is credited with the discovery of the constant ratio of the circumference of a circle to its diameter. We call it the number pi. It is irrational, that is, it cannot be expressed as an ordinary or periodic fraction. The most commonly used value for the number π is 3.14, accurate to two decimal places. The length of a circle L with radius R is easily calculated using the formula: L=2πR.

A circle of radius R=1 has a length of 2π. This relationship is used in geometry as a formulation of the radian measure of angle.

By definition, a radian is an angle with its vertex at the center of the circle, subtended by an arc with a length equal to the radius of the circle. The international designation for the radian is rad, the domestic designation is rad. It has no dimension.

An arc of a circle with radius R and angular value α radians has length α * R.

Why was it necessary to introduce a new unit of measurement for angle?

The development of science and technology led to the emergence of trigonometry and mathematical analysis, necessary for accurate calculations of mechanical and optical devices. One of its tasks is to measure the length of a curved line. The most common case is determining the length of a circular arc. Using the degree measure of angles for this purpose is extremely inconvenient. The idea of ​​comparing the length of an arc with the radius of a circle arose among many mathematicians, but the term “radian” itself was introduced into scientific use only in the second half of the 19th century. Now all trigonometric functions in mathematical analysis By default, the radian angle measure is used.

How to convert degrees to radians

From the formula for the circumference of a circle it follows that 2π radii fit into it. It follows that: 1⁰=2π/360= π/180 rad.

And a simple formula for converting from radians to degrees: 1 rad = 180/π.

Let us have an angle of N degrees. Then the formula for converting from degrees to radians will be: α(radians) = N/(180/π) = N*π/180.

Still have questions?

The answers to them can be found, where the concepts of circumference, radian measure of angles and specific examples shows the conversion of degrees to radians. Knowledge of the above is extremely important for understanding mathematics, without which the existence of modern civilization is impossible.

People in mathematical science quite often encounter such a problem as converting degrees to radians or vice versa. This task is quite simple to complete and you do not need to have deep knowledge of various applied sciences or mathematics. So, first you need to understand these measurement quantities. The degree and radian are the basic units used to measure plane angles in mathematics and physics. These units are also used in cartography to determine coordinates anywhere on the globe.

These measurement quantities are designated as follows:

• rad – radian
• degree - º

How to convert degrees to radians

To begin with, in order to understand the formula for converting degrees to radians, you need to learn how to convert an angle into radians and radians into an angle:

• 1 rad = (180/π)ºπ 57.295779513, where it is known that π = 3.14
• 1° = (π/180) rad π 0.017453293 rad

Based on the above formulas, it immediately becomes clear that π rad = 180°, and it is from them that the simple and understandable formulas for converting measurement values ​​originate. Now let's look at the basic formulas that are used in translation:

1. Degrees to radians

Zº=Z rad × (180/π), where Zº is the angle in degrees, and Z rad is the angle in radians, π = 3.14

2. Radians to degrees

Z rad = Z° × (π/180)

Now let's look at an example to make it clearer how to use the above formulas in practice. To do this, take two angles of 20º and 100º:

1. Convert degrees to radians

• 20º = 20 rad × (π/180) π 0.35 rad
• 100º = 100 rad × (180/π) π 1.7453 rad

2. Convert radians to degrees

• 20 rad = 20º × (180/π) π 1146.15, where π = 3.14
• 100 rad = 100° × (180/π) π 5729.577, where π = 3.14

Having examined the formulas for converting measurement values, it becomes clear that coping with the task is quite simple. For those people who do not want to carry out calculations on their own, there are many sites on the Internet where, using online calculators, you can convert degrees to radians or vice versa; their use will make it much easier for you to perform various trigonometry tasks.

The online calculator performs convert degrees to radians, Convert radians to degrees, conversion of fractional degrees (degrees represented decimal) in the form of degrees, minutes and seconds and displays formulas with detailed solutions.

Convert degrees to radians: degrees must be multiplied by π/180. If degrees are specified in the form of “degrees, minutes and seconds”, then first they must be converted to decimal form using the formula: degrees + minutes/60 + seconds/3600;

Formula for converting radians to degrees: if the angle is equal to α rad radians, then it is equal to formula for converting radians to degrees degrees, where π ≈ 3.1415.

Convert radians to degrees: radians must be multiplied by 180/π. The integer part of the resulting product is the number of degrees. To convert a fractional part into minutes, you need to multiply it by 60. The integer part of the resulting product is the number of minutes. To calculate seconds, you must again multiply the fractional part from the previous operation by 60, round the resulting product to the nearest integer - this is the number of seconds.

Formula for converting degrees to radians: if the angle is equal to α deg radians, then it is equal to formula for converting degrees to radians radians, where π ≈ 3.1415.

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I. Note:

1. Rounding of calculation results is performed to the specified number of decimal places (by default - rounding to ten thousandths).

II. For reference:

1. Degree measure of angle- an angular measure in which an angle of 1 degree is taken as a unit and shows how many times a degree and its parts (minute and second) fit into a given angle.
2. Radian measure of angle- an angular measure in which an angle of 1 radian is taken as a unit and shows how many times a radian fits into a given angle.
3. Degrees and radians- units of measurement of plane angles in geometry.
4. One degree equal to 1/180 of a turned angle.
5. Radian- the angle corresponding to an arc whose length is equal to its radius.

Nomogram for converting radians to degrees and degrees to radians.

Table of values trigonometric functions

Note. This table of trigonometric function values ​​uses the √ sign to indicate 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”)