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.

`cospi(x)`

, `sinpi(x)`

, and `tanpi(x)`

, compute
`cos(pi*x)`

, `sin(pi*x)`

, and `tan(pi*x)`

.

```
cos(x)
sin(x)
tan(x)
```acos(x)
asin(x)
atan(x)
atan2(y, x)

cospi(x)
sinpi(x)
tanpi(x)

x, y

numeric or complex vectors.

`tanpi(0.5)`

is `NaN`

. Similarly for other inputs
with fractional part `0.5`

.

For the inverse trigonometric functions, branch cuts are defined as in Abramowitz and Stegun, figure 4.4, page 79.

For `asin`

and `acos`

, there are two cuts, both along
the real axis: \(\left(-\infty, -1\right]\) and
\(\left[1, \infty\right)\).

For `atan`

there are two cuts, both along the pure imaginary
axis: \(\left(-\infty i, -1i\right]\) and
\(\left[1i, \infty i\right)\).

The behaviour actually on the cuts follows the C99 standard which requires continuity coming round the endpoint in a counter-clockwise direction.

Complex arguments for `cospi`

, `sinpi`

, and `tanpi`

are not yet implemented, and they are a ‘future direction’ of
ISO/IEC TS 18661-4.

All except `atan2`

are S4 generic functions: methods can be defined
for them individually or via the
`Math`

group generic.

The arc-tangent of two arguments `atan2(y, x)`

returns the angle
between the x-axis and the vector from the origin to \((x, y)\),
i.e., for positive arguments `atan2(y, x) == atan(y/x)`

.

Angles are in radians, not degrees, for the standard versions (i.e., a
right angle is \(\pi/2\)), and in ‘half-rotations’ for
`cospi`

etc.

`cospi(x)`

, `sinpi(x)`

, and `tanpi(x)`

are accurate
for `x`

values which are multiples of a half.

All except `atan2`

are internal generic primitive
functions: methods can be defined for them individually or via the
`Math`

group generic.

These are all wrappers to system calls of the same name (with prefix
`c`

for complex arguments) where available. (`cospi`

,
`sinpi`

, and `tanpi`

are part of a C11 extension
and provided by e.g.macOS and Solaris: where not yet
available call to `cos`

*etc* are used, with special cases
for multiples of a half.)

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
*The New S Language*.
Wadsworth & Brooks/Cole.

Abramowitz, M. and Stegun, I. A. (1972). *Handbook of
Mathematical Functions*. New York: Dover.
Chapter 4. Elementary Transcendental Functions: Logarithmic,
Exponential, Circular and Hyperbolic Functions

For `cospi`

, `sinpi`

, and `tanpi`

the C11 extension
ISO/IEC TS 18661-4:2015 (draft at
http://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf).

```
# NOT RUN {
x <- seq(-3, 7, by = 1/8)
tx <- cbind(x, cos(pi*x), cospi(x), sin(pi*x), sinpi(x),
tan(pi*x), tanpi(x), deparse.level=2)
op <- options(digits = 4, width = 90) # for nice formatting
head(tx)
tx[ (x %% 1) %in% c(0, 0.5) ,]
options(op)
# }
```

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