asin

Symbolic inverse sine function

Syntax

asin(X)

Description

asin(X) returns the inverse sine function (arcsine function) of X. All angles are in radians.

• For real values of X in the interval [-1,1], asin(X) returns the values in the interval [-pi/2,pi/2].

• For real values of X outside the interval [-1,1] and for complex values of X, asin(X) returns complex values with the real parts in the interval [-pi/2,pi/2].

Examples

Inverse Sine Function for Numeric and Symbolic Arguments

Depending on its arguments, asin returns floating-point or exact symbolic results.

Compute the inverse sine function for these numbers. Because these numbers are not symbolic objects, asin returns floating-point results.

A = asin([-1, -1/3, -1/2, 1/4, 1/2, sqrt(3)/2, 1])

A =
    -1.5708   -0.3398   -0.5236    0.2527    0.5236    1.0472    1.5708

Compute the inverse sine function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, asin returns unresolved symbolic calls.

symA = asin(sym([-1, -1/3, -1/2, 1/4, 1/2, sqrt(3)/2, 1]))

symA =
[ -pi/2, -asin(1/3), -pi/6, asin(1/4), pi/6, pi/3, pi/2]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)

ans =
[ -1.5707963267948966192313216916398,...
-0.33983690945412193709639251339176,...
-0.52359877559829887307710723054658,...
0.25268025514207865348565743699371,...
0.52359877559829887307710723054658,...
1.0471975511965977461542144610932,...
1.5707963267948966192313216916398]

Plot Inverse Sine Function

Plot the inverse sine function on the interval from -1 to 1.

syms x
fplot(asin(x),[-1 1])
grid on

Handle Expressions Containing Inverse Sine Function

Many functions, such as diff, int, taylor, and rewrite, can handle expressions containing asin.

Find the first and second derivatives of the inverse sine function:

syms x
diff(asin(x), x)
diff(asin(x), x, x)

ans =
1/(1 - x^2)^(1/2)
 
ans =
x/(1 - x^2)^(3/2)

Find the indefinite integral of the inverse sine function:

int(asin(x), x)

ans =
x*asin(x) + (1 - x^2)^(1/2)

Find the Taylor series expansion of asin(x):

taylor(asin(x), x)

ans =
(3*x^5)/40 + x^3/6 + x

Rewrite the inverse sine function in terms of the natural logarithm:

rewrite(asin(x), 'log')

ans =
-log((1 - x^2)^(1/2) + x*1i)*1i

Input Arguments

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.