Trigonometric Functions

Description

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).

Usage

cos(x)
sin(x)
tan(x)

acos(x)
asin(x)
atan(x)
atan2(y, x)

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

Arguments

Details

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.)

Value

tanpi(0.5) is NaN. Similarly for other inputs with fractional part 0.5.

Complex values

For the inverse trigonometric functions, branch cuts are defined as in .

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.

S4 methods

All except atan2 are S4 generic functions: methods can be defined for them individually or via the Math group generic.

References

⁠Becker RA, Chambers JM, Wilks AR (1988). The New S Language. Chapman and Hall/CRC, London.

For cospi, sinpi, and tanpi the C11 extension ISO/IEC TS 18661-4:2015 (draft at https://www.open-std.org/jtc1/sc22/wg14/www/docs/n1950.pdf).

Examples

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)