Editing Astronomical coordinate systems (section)

=== Equatorial ↔ ecliptic ===
The classical equations, derived from [[spherical trigonometry]], for the longitudinal coordinate are presented to the right of a bracket; dividing the first equation by the second gives the convenient tangent equation seen on the left.<ref name=ExplSupp>
{{cite book 
 | last1 = U.S. Naval Observatory
 | first1=Nautical Almanac Office
 | last2 = H.M. Nautical Almanac Office
 | title = Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac
 | publisher = H.M. Stationery Office, London
 | year = 1961
}}, sec. 2A</ref>  The rotation matrix equivalent is given beneath each case.<ref>
{{cite book 
 | last1 = U.S. Naval Observatory
 | first1=Nautical Almanac Office
 | editor = P. Kenneth Seidelmann
 | title = Explanatory Supplement to the Astronomical Almanac
 | publisher = University Science Books, Mill Valley, CA
 | year = 1992
 | isbn = 0-935702-68-7
}}, section 11.43</ref> This division is ambiguous because tan has a period of 180° ({{pi}}) whereas cos and sin have periods of 360° (2{{pi}}).

:<math>\begin{align}
   \tan\left(\lambda\right) &= {\sin\left(\alpha\right) \cos\left(\varepsilon\right) + \tan\left(\delta\right) \sin\left(\varepsilon\right) \over \cos\left(\alpha\right)}; \qquad\begin{cases}
    \cos\left(\beta\right) \sin\left(\lambda\right) = \cos\left(\delta\right) \sin\left(\alpha\right) \cos\left(\varepsilon\right) + \sin\left(\delta\right) \sin\left(\varepsilon\right); \\
    \cos\left(\beta\right) \cos\left(\lambda\right) = \cos\left(\delta\right) \cos\left(\alpha\right).
  \end{cases} \\
     \sin\left(\beta\right) &= \sin\left(\delta\right) \cos\left(\varepsilon\right) - \cos\left(\delta\right) \sin\left(\varepsilon\right) \sin\left(\alpha\right) \\[3pt]
  \begin{bmatrix}
    \cos\left(\beta\right)\cos\left(\lambda\right) \\
    \cos\left(\beta\right)\sin\left(\lambda\right) \\
    \sin\left(\beta\right)
              \end{bmatrix} &= \begin{bmatrix}
    1 &  0                            & 0                            \\
    0 &  \cos\left(\varepsilon\right) & \sin\left(\varepsilon\right) \\
    0 & -\sin\left(\varepsilon\right) & \cos\left(\varepsilon\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(\delta\right)\cos\left(\alpha\right) \\
    \cos\left(\delta\right)\sin\left(\alpha\right) \\
    \sin\left(\delta\right)
  \end{bmatrix} \\[6pt]
    \tan\left(\alpha\right) &= {\sin\left(\lambda\right) \cos\left(\varepsilon\right) - \tan\left(\beta\right) \sin\left(\varepsilon\right) \over \cos\left(\lambda\right)} ; \qquad \begin{cases}
    \cos\left(\delta\right) \sin\left(\alpha\right) = \cos\left(\beta\right) \sin\left(\lambda\right) \cos\left(\varepsilon\right) - \sin\left(\beta\right) \sin\left(\varepsilon\right); \\
    \cos\left(\delta\right) \cos\left(\alpha\right) = \cos\left(\beta\right) \cos\left(\lambda\right).
  \end{cases} \\[3pt]
    \sin\left(\delta\right) &= \sin\left(\beta\right) \cos\left(\varepsilon\right) + \cos\left(\beta\right) \sin\left(\varepsilon\right) \sin\left(\lambda\right). \\[6pt]
  \begin{bmatrix}
    \cos\left(\delta\right)\cos\left(\alpha\right) \\
    \cos\left(\delta\right)\sin\left(\alpha\right) \\
    \sin\left(\delta\right)
  \end{bmatrix} &= \begin{bmatrix}
    1 & 0                            &  0                            \\
    0 & \cos\left(\varepsilon\right) & -\sin\left(\varepsilon\right) \\
    0 & \sin\left(\varepsilon\right) &  \cos\left(\varepsilon\right)
  \end{bmatrix}\begin{bmatrix}
    \cos\left(\beta\right)\cos\left(\lambda\right) \\
    \cos\left(\beta\right)\sin\left(\lambda\right) \\
    \sin\left(\beta\right)
  \end{bmatrix}.
\end{align}</math>