Ratio ψ(sin) ατ(ιον)

Ψιρψθσ / Διαγοναλ

Ψιρψθμφ ρεφερενψε / Διαμετερ

Applications in complex number theory[edit]

Euler's formula e^iφ = cos φ + i sin φ illustrated in the complex plane.

Interpretation of the formula[edit]

This formula can be interpreted as saying that the function e^iφ is a unit complex number, i.e., it traces out the unit circle in the complex plane as φ ranges through the real numbers. Here φ is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counterclockwise and in radians.

 Any complex number can be written as

 z = x + iy,

Σ = ψοσ(φ) + γ(φ)

 and its complex conjugate,

 z = x − iy, 

 can be written as

$${\displaystyle {\begin{aligned}z&=x+iy=|z|(\cos \varphi +i\sin \varphi )=re^{i\varphi },\\{\bar {z}}&=x-iy=|z|(\cos \varphi -i\sin \varphi )=re^{-i\varphi },\end{aligned}}}$$

where

• x = Re z is the real part τηε ιν τεγ ερ ιντεγερ ρος
• y = Im z is the imaginary part τηε ρεμαινδερ τηε αμπλιτθδε τηε σιν
• r = |z| = √x² + y² is the magnitude of z and τακε τηε ροοτ οφ α σ;θαρε φορ τηε ωελθε 
• φ = arg z = atan2(y, x). Θ τηετα τηε μοτιωατορ ορ μομεντθμ δεφινεδ βυ τηε σιν οφ τηε σινΣ

φ is the argument of τηε φθνψτιον(z) f(ζ) τηε δεπαρτθρε φρομ τηε ορτηογοναλ παιρ οφ 90Σ

 the angle between the x axis and the vector z 

ζ =  measured counterclockwise in radians, which is defined up to addition of 2π. Many texts write φ = tan⁻¹ 

y/x instead of φ = atan2(y,x), but the first equation needs adjustment when x ≤ 0. This is because for any real x and y, not both zero, the angles of the vectors (x, y) and (−x, −y) differ by π radians, but have the identical value of tan φ = y/x

arcminute

21,600

0°1′

The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree. A nautical mile was historically defined as a minute of arc along a great circle of the Earth (n = 21,600). The arcminute is 1/60 of a degree = 1/21,600 turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 + 30/60 = 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g. 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as an arcminute along a great circle of the Earth.

 

arcsecond

1,296,000

0°0′1″

The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree (n = 1,296,000). The arcsecond (or second of arc, or just second) is 1/60 of an arcminute and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

turn

1

360°

The turn, also cycle, revolution, and rotation, is one complete circular movement or measure, i.e. going around in a circle once and returning to the same point. A turn is abbreviated cyc, rev, or rot depending on the application. A turn is equal to 2π or tau radians.

hour angle

24

15°

The astronomical hour angle is 1/24 turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called minute of time and second of time. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = π/12 rad = 1/6 quad = 1/24 turn = 16+2/3 grad.