acos

acos(x: array, /) → array

Calculates an implementation-dependent approximation of the principal value of the inverse cosine for each element x_i of the input array x.

Each element-wise result is expressed in radians.

Note

The principal value of the arc cosine of a complex number \(z\) is

\[\operatorname{acos}(z) = \frac{1}{2}\pi + j\ \ln(zj + \sqrt{1-z^2})\]

For any \(z\),

\[\operatorname{acos}(z) = \pi - \operatorname{acos}(-z)\]

Note

For complex floating-point operands, acos(conj(x)) must equal conj(acos(x)).

Note

The inverse cosine (or arc cosine) is a multi-valued function and requires a branch cut on the complex plane. By convention, a branch cut is placed at the line segments \((-\infty, -1)\) and \((1, \infty)\) of the real axis.

Accordingly, for complex arguments, the function returns the inverse cosine in the range of a strip unbounded along the imaginary axis and in the interval \([0, \pi]\) along the real axis.

Note: branch cuts follow C99 and have provisional status (see Branch Cuts).

Parameters:

x (array) – input array. Should have a floating-point data type.

Returns:

out (array) – an array containing the inverse cosine of each element in x. The returned array must have a floating-point data type determined by Type Promotion Rules.

Notes

Special cases

For real-valued floating-point operands,

  • If x_i is NaN, the result is NaN.

  • If x_i is greater than 1, the result is NaN.

  • If x_i is less than -1, the result is NaN.

  • If x_i is 1, the result is +0.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

  • If a is either +0 or -0 and b is +0, the result is π/2 - 0j.

  • If a is either +0 or -0 and b is NaN, the result is π/2 + NaN j.

  • If a is a finite number and b is +infinity, the result is π/2 - infinity j.

  • If a is a nonzero finite number and b is NaN, the result is NaN + NaN j.

  • If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is π - infinity j.

  • If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +0 - infinity j.

  • If a is -infinity and b is +infinity, the result is 3π/4 - infinity j.

  • If a is +infinity and b is +infinity, the result is π/4 - infinity j.

  • If a is either +infinity or -infinity and b is NaN, the result is NaN ± infinity j (sign of the imaginary component is unspecified).

  • If a is NaN and b is a finite number, the result is NaN + NaN j.

  • If a is NaN and b is +infinity, the result is NaN - infinity j.

  • If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.