\HeaderA{Trig}{Trigonometric Functions}{Trig}
\aliasA{acos}{Trig}{acos}
\aliasA{asin}{Trig}{asin}
\aliasA{atan}{Trig}{atan}
\aliasA{atan2}{Trig}{atan2}
\aliasA{cos}{Trig}{cos}
\aliasA{sin}{Trig}{sin}
\aliasA{tan}{Trig}{tan}
\keyword{math}{Trig}
\begin{Description}\relax
These functions give the obvious trigonometric functions.  They
respectively compute the cosine, sine, tangent, arc-cosine, arc-sine,
arc-tangent, and the two-argument arc-tangent.
\end{Description}
\begin{Usage}
\begin{verbatim}
cos(x)
sin(x)
tan(x)
acos(x)
asin(x)
atan(x)
atan2(y, x)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x, y}] numeric or complex vectors.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
The arc-tangent of two arguments \code{atan2(y, x)} returns the angle
between the x-axis and the vector from the origin to \eqn{(x, y)}{},
i.e., for positive arguments \code{atan2(y, x) == atan(y/x)}.

Angles are in radians, not degrees (i.e., a right angle is
\eqn{\pi/2}{}).

All except \code{atan2} are generic functions: methods can be defined
for them individually or via the \code{\LinkA{Math}{Math}}
group generic.
\end{Details}
\begin{Section}{Complex values}
For the inverse trigonometric functions, branch cuts are defined as in
Abramowitz and Stegun, figure 4.4, page 79.  Continuity on the
branch cuts is standard.

For \code{asin()} and \code{acos()}, there are two cuts, both along
the real axis: \eqn{\left(-\infty, -1\right]}{\(-Inf, -1\]} and
\eqn{\left[1, \infty\right)}{\[1, Inf\)}.  Functions \code{asin()}
and \code{acos()} are continuous from above on the interval
\eqn{\left(-\infty, -1\right]}{\(-Inf, -1\]} and continuous from
below on \eqn{\left[1, \infty\right)}{\[1, Inf\)}.

For \code{atan()} there are two cuts, both along the pure imaginary
axis: \eqn{\left(-\infty i, -1i\right]}{\(-1i*Inf, -1i\]} and
\eqn{\left[1i, \infty i\right)}{\[1i, 1i*Inf\)}.  It is continuous
from the left on the interval
\eqn{\left(-\infty i, -1i\right]}{\(-1i*Inf, -1i\]} and from the
right on the interval \eqn{\left[1i, \infty i\right)}{\[1i, 1i*Inf\)}.
\end{Section}
\begin{Section}{S4 methods}
All except \code{atan2} are S4 generic functions: methods can be defined
for them individually or via the
\code{\LinkA{Math}{Math}} group generic.
\end{Section}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole.

Abramowitz, M. and Stegun, I. A. (1972). \emph{Handbook of
Mathematical Functions,} New York: Dover.\\
Chapter 4. Elementary Transcendental Functions: Logarithmic,
Exponential, Circular and Hyperbolic Functions
\end{References}

