Angular unit in the context of "Turn (geometry)"

A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol ′, is a unit of angular measurement equal to ⁠1/60⁠ of a degree. Since one degree is ⁠1/360⁠ of a turn, or complete rotation, one arcminute is ⁠1/21600⁠ of a turn. The nautical mile (nmi) was originally defined as the arc length of a minute of latitude on a spherical Earth, so the actual Earth's circumference is very near 21600 nmi. A minute of arc is ⁠π/10800⁠ of a radian.

A second of arc, arcsecond (abbreviated as arcsec), or arc second, denoted by the symbol ″, is a unit of angular measurement equal to ⁠1/60⁠ of a minute of arc, ⁠1/3600⁠ of a degree, ⁠1/1296000⁠ of a turn, and ⁠π/648000⁠ (about ⁠1/206264.8⁠) of a radian.

In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function ${\displaystyle F}$ of some real variable ${\displaystyle t}$ (such as time) is an angle-like quantity representing the fraction of the cycle covered up to ${\displaystyle t}$. It is expressed in such a scale that it varies by one full turn as the variable ${\displaystyle t}$ goes through each period (and ${\displaystyle F(t)}$ goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or ${\displaystyle 2\pi }$ as the variable ${\displaystyle t}$ completes a full period.

This convention is especially appropriate for a sinusoidal function, since its value at any argument ${\displaystyle t}$ then can be expressed as ${\displaystyle \varphi (t)}$, the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.)

The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to 2π radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (symbol rev or r). Common related units of frequency are cycles per second (cps) and revolutions per minute (rpm). The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc.

In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless quantity called rotation, defined as the ratio of a given angle and a full turn. It is represented by the symbol N. (See below for the formula.)