4.8.2 Trigonometric functions

cl_R sin (const cl_R& x)

Returns sin(x). The range of the result is the interval -1 <= sin(x) <= 1.

cl_N sin (const cl_N& z)

Returns sin(z). The range of the result is the entire complex plane.

cl_R cos (const cl_R& x)

Returns cos(x). The range of the result is the interval -1 <= cos(x) <= 1.

cl_N cos (const cl_N& x)

Returns cos(z). The range of the result is the entire complex plane.

struct cos_sin_t { cl_R cos; cl_R sin; };

cos_sin_t cos_sin (const cl_R& x)

Returns both sin(x) and cos(x). This is more efficient than computing them separately. The relation cos^2 + sin^2 = 1 will hold only approximately.

cl_R tan (const cl_R& x)

cl_N tan (const cl_N& x)

Returns tan(x) = sin(x)/cos(x).

cl_N cis (const cl_R& x)

cl_N cis (const cl_N& x)

Returns exp(i*x). The name ‘cis’ means “cos + i sin”, because e^(i*x) = cos(x) + i*sin(x).

cl_N asin (const cl_N& z)

Returns arcsin(z). This is defined as arcsin(z) = log(iz+sqrt(1-z^2))/i and satisfies arcsin(-z) = -arcsin(z). The range of the result is the strip in the complex domain -pi/2 <= realpart(arcsin(z)) <= pi/2, excluding the numbers with realpart = -pi/2 and imagpart < 0 and the numbers with realpart = pi/2 and imagpart > 0.

cl_N acos (const cl_N& z)

Returns arccos(z). This is defined as arccos(z) = pi/2 - arcsin(z) = log(z+i*sqrt(1-z^2))/i and satisfies arccos(-z) = pi - arccos(z). The range of the result is the strip in the complex domain 0 <= realpart(arcsin(z)) <= pi, excluding the numbers with realpart = 0 and imagpart < 0 and the numbers with realpart = pi and imagpart > 0.

cl_R atan (const cl_R& x, const cl_R& y)

Returns the angle of the polar representation of the complex number x+iy. This is atan(y/x) if x>0. The range of the result is the interval -pi < atan(x,y) <= pi. The result will be an exact number only if x > 0 and y is the exact 0. WARNING: In Common Lisp, this function is called as (atan y x), with reversed order of arguments.

cl_R atan (const cl_R& x)

Returns arctan(x). This is the same as atan(1,x). The range of the result is the interval -pi/2 < atan(x) < pi/2. The result will be an exact number only if x is the exact 0.

cl_N atan (const cl_N& z)

Returns arctan(z). This is defined as arctan(z) = (log(1+iz)-log(1-iz)) / 2i and satisfies arctan(-z) = -arctan(z). The range of the result is the strip in the complex domain -pi/2 <= realpart(arctan(z)) <= pi/2, excluding the numbers with realpart = -pi/2 and imagpart >= 0 and the numbers with realpart = pi/2 and imagpart <= 0.

Archimedes’ constant pi = 3.14… is returned by the following functions:

cl_F pi (float_format_t f)

Returns pi as a float of format f.

cl_F pi (const cl_F& y)

Returns pi in the float format of y.

cl_F pi (void)

Returns pi as a float of format default_float_format.

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