What is a radian? Definition of radian

What is a radian
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Radian is a unit of angle called the arc degree method. Introduced in the second year of high school with the of trigonometric functions, it is used mainly in fields such as mathematics and physics.

This page explains the meaning of radians and how to convert them to degrees in degrees.

What is radian

Definition of radian

The radian is the unit of plane angle of the International System of Units (SI) and is defined as follows.

1 Radian is the magnitude of the central angle for an arc equal to the length of the radius of the circle

The symbol of the unit is rad, and the expression of the angle with this as the unit is called the radian method.

Even so, I think it’s hard to get an image just by listening to it once. Therefore, let’s understand the definition using figures.

In the figure below, an orange line shows an arc with a length equal to the radius of the circle. The magnitude of the central angle (orange corner) for this arc is 1 radian.

1 radian diagram(1 radian≈57.2958°)

As explained in the next conversion section, converting one radian to “degrees (°)” results in a half-hearted number of about 57.2958°. In fact, this value is an infinite number of decimals.

This was a problem! You may think that it is a difficult way to express an angle, but don’t worry. Whereas the degree method sometimes uses 1 °, 2 °,…, it is unlikely that radians are separated by integer values ​​such as 1 rad, 2 rad,…. For radians, the basic π is used to represent the angle.

Next, I will explain the conversion between radians and “degrees”.

By the way, radians are often not given the unit rad. The reason is that the radians are “the ratio of the length of the arc to the length of the radius”, that is, “the length divided by the length”, so the units cancel each other out.

How to use radians

In fact, there is no merit in using radians, considering only the ratio of r, x, y to the angle using trigonometric functions .

It is used for linear approximation of angles (simplification and speeding up of calculations), but the meaning of being radians is not so strong.

Radians come into their own when differentiation is involved.

When trigonometric functions are involved in differentiation, the story does not work at all in “degrees“, and “radians” are the main.

The true use of radians will be explained when explaining the differentiation of trigonometric functions.

The reason why we are talking about radians now is that the angle value input to trigonometric functions when dealing with trigonometric functions on a computer is in radians. Trigonometric functions are in radians not only in C language but also in Excel. Exceptionally, Unity of the game engine is in units of degrees, but since the program is written in C # and Math is used for calculation, it is handled in units of radians in the end. I think there are various reasons why it is radian when dealing with trigonometric functions on a computer, but one of the major reasons is that series expansion (polynomial approximation) is performed to obtain highly accurate calculation results.

So, of course, there is no need to learn differentiation and series expansion to handle trigonometric functions. In order to understand trigonometric functions deeply, it is necessary to understand differentiation, and in order to understand internal processing, it is necessary to understand series expansion, but when dealing with trigonometric functions for the first time, it is sufficient if the degrees can be converted to radians.

For the time being, when inputting an angle to a trigonometric function on a computer, it is necessary to use it in radians, so convert it to
 angle [rad] = angle [deg] / 180 * PI
before inputting.

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