Degrees ⚠️ in radians: how to translate, formula, conversion table

Degrees in radians

Short description

Angle - These are two beams emerging from one point. This point is called a vertex. Taking a specific angle for a unit of measurement, it is possible to determine the value of any angle, finding out how many times the unit angle is stacked. When measuring the angle, it is based on its properties:

  1. The values ​​of equal corners are equal.
  2. The amount of the sum of two angles is equal to the sum of their values.

If it is clear what we are talking about, instead of the "corner magnitude" they say simply "angle".

Equal angles with a vertex in the center of the circle will create arcs on it the same length. Their amount will be equal to the sum of the arcs tightened. Therefore, the units of measurement of angles can be set, indicating which part of the circle is the corresponding arc.

What can measure angle

Two units of measure are the most common:

  • Degree, equal to an arc in 1/360 of the whole circle;
  • Radine is a central angle carving an arc equal to the circle radius.

There are still such units of measure, as a hail equal to 1/100 direct angle, a turnover equal to a complete circle, thousandth - \ (\ FRAC1 {2 \ Mathrm \ Pi \; \ Times \; 1000} \) and Rumbers - 1/32 full circumference.

Communication between degrees and radians

Measure corner

In practice, degrees are most often used. They are denoted by the sign \ (^ \ Circ \; \) .

1/60 degrees - a minute denoted by the sign '. For a second designate the sign ", it is 1/3600 lobes.

Mathematics and astronomers prefer to use radians, dimensionless magnitude. It is more convenient when considering trigonometric functions. The designation "Rad" is usually lowered. Radian is approximately about \ (57 ^ \ CIRC17'45 '' \) .

Relationship formula

The length of the arc dried in A radian on the circle of the radius R, is calculated by multiplying A to R, and for a single circle, the length of the arc and the angle value coincides.

Since the radius is equal to one, the length of the unit circle will be equal to \ (2 \ mathrm \ pi \) .

Thus, the connection of radians and degrees can be expressed by the formula

\ (1 \; \ Mathrm {radian} \; = \; \ FRAC1 {\ left (2 \ mathrm \ pi \ right)} \; \ mathrm {revolutions} \; = \; \ FRAC {180} {\ Mathrm \ pi} \; \ mathrm {degrees} \; = \; \ frac {200} {\ mathrm \ pi} \; \ mathrm {hav} \)

Formulas transfer

Degrees in radians

Carry out the transition from degrees to radians by the formula

\ (\ MathRM x ^ \ Circ \; = \; \ FRAC {\ PI \; \ Times \; A \; Rad} {180 ^ \ CIRC} \)

Radians in degrees

\ (A \; Rad \; = \; x ^ \ Circ \; \ Times \; (\; \ FRAC {\ Mathrm \ pi} {180 ^ \ Circ}) \)

Degree Degree Table in Radian

The ratio of two measurement systems on the circle can be visually seen in the diagram:

Radians and degrees on circle

But in practice, to convert one dimension to another to another, it is more convenient to use the table:

Radian table and degrees

Examples of calculation of degrees and minutes in radians

Example 1.

Translate 35 degrees to radians.

Decision

According to the formula, we need to be 35 multiplied by the PI number and divided by 180.

\ (x \; rad \; = \; 35 ^ \ Circ \; \ Times \; (\; \ FRAC {\ MathRM \ pi} {180 ^ \ CIRC} \; \ approx \; 35 ^ \ CIRC \ ; \ Times \; (\; \ FRAC {3,1416} {180 ^ \ CIRC}) \; \ APPROX \; 0,6108 \\ 35 ^ \ Circ \; = \; 0,6108 \; Rad \)

To translate minutes and seconds, you must first translate them to degrees.

Example 2.

Translate into radians angle \ (87 ^ \ CIRC14'21 '' \) .

Decision

\ (1 ^ \ Circ \; = \; 60 '\; = \; 3600' '\)

\ (87 ^ \ CIRC + \; \ FRAC {14 '} {60} \; + \; \ FRAC {21' '} {3600} \; = \; 87 ^ \ Circ \; + \; 0,2333 ^ \ Circ \; + \; \; 0.0058 ^ \ CIRC \; = \; 87,239 ^ \ CIRC \)

We use the formula, substituting the value found:

\ (x \; rad \; = \; 87,239 ^ \ Circ \; \ Times \; (\; \ FRAC {\ Mathrm \ pi} {180 ^ \ Circ}) \; \ approx \; 87,239 ^ \ CIRC \ ; \ Times \; (\; \ FRAC {3,1416} {180 ^ \ CIRC}) \; \ approx \; 1,5226 \\ 87,239 ^ \ CIRC \; = 1,5226 \; Rad \)

Example 3.

Translate into radians angle \ (194 ^ \ CIRC19 '\) .

Decision

\ (1 ^ \ Circ \; = \; 60 '\; = \; 3600' '\)

\ (194 ^ \ CIRC + \; \ FRAC {19 '} {60} \; = \; 194 ^ \ Circ \; + \; 0.3166 ^ \ Circ \; = \; 194,3166 ^ \ CIRC \)

\ (x \; rad \; = \; 194,3166 ^ \ CIRC \; \ Times \; (\; \ FRAC {\ Mathrm \ pi} {180 ^ \ Circ}) \; \ Approx \; 194,3166 ^ \ Circ \; \ Times \; (\; \ FRAC {3,1416} {180 ^ \ CIRC}) \; \ approx \; 3,3915 \\ 194,3166 ^ \ CIRC \; = 3,3915 \ ;glad\)

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Translation of degrees in radians (π) online and vice versa

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Instructions for calculator

  • Enter degrees or radians with π(PI) in the field above and click the "Translate" button.

Translate radians and degrees online

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