Inverse sine function

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.




arcsin(x) represents the inverse of the sine function.

The angle returned by this function is measured in radians, not in degrees. For example, the result π represents an angle of 180o.

arcsin is defined for complex arguments.

Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for interval arguments. Unevaluated function calls are returned for most exact arguments.

If the argument is a rational multiple of I, the result is expressed in terms of hyperbolic functions. See Example 2.

The inverse sine function is multivalued. The MuPAD® arcsin function returns the value on the main branch. The branch cuts are the real intervals (- ∞, - 1) and (1, ∞). Thus, arcsin returns values, such that y = arcsin(x) satisfies π2(y)π2 for any finite complex x.

The sin function returns explicit values for arguments that are certain rational multiples of π. For these values, arcsin returns an appropriate rational multiple of π on the main branch. See Example 3.

The values jump when the arguments cross a branch cut. See Example 4.

The float attributes are kernel functions. Thus, floating-point evaluation is fast.

Environment Interactions

When called with a floating-point argument, arcsin is sensitive to the environment variable DIGITS which determines the numerical working precision.


Example 1

Call arcsin with the following exact and symbolic input arguments:

arcsin(1), arcsin(1/sqrt(2)), arcsin(5 + I),
arcsin(1/3), arcsin(I), arcsin(sqrt(2))

arcsin(-x), arcsin(x + 1), arcsin(1/x)

Floating-point values are computed for floating-point arguments:

arcsin(0.1234), arcsin(5.6 + 7.8*I), arcsin(1.0/10^20)

Floating-point intervals are computed for interval arguments:

arcsin(-1/2...1/2), arcsin(0...1)

Example 2

Arguments that are rational multiples of I are rewritten in terms of hyperbolic functions:

arcsin(5*I), arcsin(5/4*I), arcsin(-3*I)

For other complex arguments unevaluated function calls without simplifications are returned:

arcsin(1/2^(1/2) + I), arcsin(1 - 3*I)

Example 3

Some special values are implemented:

arcsin(1/sqrt(2)), arcsin((5^(1/2) - 1)/4), arcsin(3^(1/2)/2)

Such simplifications occur for arguments that are trigonometric images of rational multiples of π:

sin(9/10*PI), arcsin(sin(9/10*PI))

Example 4

The values jump when crossing a branch cut:

arcsin(2.0 + I/10^10), arcsin(2.0 - I/10^10)

On the branch cut, the values of arcsin coincide with the limit "from below" for real arguments x > 1:

limit(arcsin(2.0 - I/n), n = infinity);
limit(arcsin(2.0 + I/n), n = infinity);

The values coincide with the limit "from above" for real x < - 1:

limit(arcsin(-2.0 - I/n), n = infinity);
limit(arcsin(-2.0 + I/n), n = infinity);

Example 5

The inverse sine function can be rewritten in terms of the logarithm function with complex arguments:

rewrite(arcsin(x), ln)

Example 6

diff, float, limit, or taylor, series, and other system functions handle expressions involving the inverse trigonometric functions:

diff(arcsin(x^2), x), float(arcsin(3)*arctan(5 + I))

limit(arcsin(1 + sin(x)/x), x = PI)

taylor(arcsin(x), x = 0)

series(arcsin(2 + x), x, 3)

Return Values

Arithmetical expression or floating-point interval.

Overloaded By


See Also

MuPAD Functions

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