In geometry, an angle is formed by two lines that meet at a point. Each line is called a side of the angle, and the point they share is called the vertex of the angle. The term angle is used to denote both geometric figures and their size or magnitude as associated quantity. Angular measure or measure of angle are sometimes used to distinguish between the measure of the quantity and figure itself. The measurement of angles is intrinsically linked with circles and rotation, and this is often visualized or defined using the arc of a circle centered at the vertex and lying between the sides.
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π Angular unit in the context of Minute of arc
A minute of arc, arcminute (abbreviated as arcmin), arc minute, or minute arc, denoted by the symbol β², is a unit of angular measurement equal to β 1/60β of a degree. Since one degree is β 1/360β of a turn, or complete rotation, one arcminute is β 1/21600β 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 21600Β nmi. A minute of arc is β Ο/10800β of a radian.
A second of arc, arcsecond (abbreviated as arcsec), or arc second, denoted by the symbol β³, is a unit of angular measurement equal to β 1/60β of a minute of arc, β 1/3600β of a degree, β 1/1296000β of a turn, and β Ο/648000β (about β 1/206264.8β ) of a radian.
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Angular unit in the context of Phase shift
In physics and mathematics, the phase (symbol Ο or Ο) of a wave or other periodic function of some real variable (such as time) is an angle-like quantity representing the fraction of the cycle covered up to . It is expressed in such a scale that it varies by one full turn as the variable goes through each period (and goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360Β° or as the variable completes a full period.
This convention is especially appropriate for a sinusoidal function, since its value at any argument then can be expressed as , 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.)
Angular unit in the context of Turn (unit)
The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angleβthe angle subtended by a complete circle at its center. One turn is equal to 2ΟΒ radians, 360Β degrees or 400Β 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.