Arcsine function

arcsin(x), sin^-1(x), inverse sine function.

The arcsine of x is defined as the inverse sine function of x when -1≤x≤1.

When the sine of y is equal to x:

sin y = x

Then the arcsine of x is equal to the inverse sine function of x, which is equal to y:

arcsin x = sin^-1 x = y

arcsin 1 = sin^-1 1 = π/2 rad = 90°

Rule name	Rule
Sine of arcsine	sin( arcsin x ) = x
Arcsine of sine	arcsin( sin x ) = x+2kπ, when k∈ℤ (k is integer)
Arcsin of negative argument	arcsin(-x) = - arcsin x
Complementary angles	arcsin x = π/2 - arccos x = 90° - arccos x
Arcsin sum	arcsin α + arcsin(β) = arcsin( α√(1-β²) + β√(1-α²) )
Arcsin difference	arcsin α - arcsin(β) = arcsin( α√(1-β²) - β√(1-α²) )
Cosine of arcsine
Tangent of arcsine
Derivative of arcsine
Indefinite integral of arcsine

See also