The function cos(arcsin x) can be used to determine the cosine of the side of a right triangle when only the angles surrounding it are provided. For example, for a right triangle with sides a, b and c where A is angle between sides a and c, the cos(arcsin(b/c)) is equal to the cos(A) because A is equal to arcsin(b/c).
The function for cos(arcsin x) is equal to sin(arccos x). Both are equal to the square root of one minus x squared. For the triangle noted above, the cos(arcsin(b/c)) is equal to a/c, which in turn can be expressed as the square root of c squared minus b squared, all divided by c. That equation is equal to the square root one minus b squared over c squared, which is then equal to the square root of sine squared of the angle of A.
The function arcsin is the inverse trigonometric function of sine, meaning that the expression x = sin y has the inverse function of y = arcsin x, which can be solved for all values equal to or less than one and equal to or greater than negative one. In radians, y must be equal to or between negative pi over two and pi over 2, which translates to negative 90 degrees and 90 degrees.