Definition

1. Noun:
   • The inverse function of the sine: arcsin is the mathematical function that, for a given number (the sine value), returns the corresponding angle. It is the function that reverses, or "undoes," the sine function.

Usage

• is used to find an angle when its sine value is known.
• It is typically written as or , where is a number between -1 and 1 inclusive.
• The result (the angle) is usually expressed in radians or degrees.

Examples

• Noun:
  • To find the angle whose sine is 0.5, you calculate arcsin(0.5).
  • The value of arcsin(1) is π/2 radians, or 90 degrees.
  • The domain of the arcsin function is the closed interval [-1, 1].

Advanced Usage

• Principal Value: The function is defined to return the principal value, which is the angle in the range [-π/2, π/2] radians (or [-90°, 90°]). This restriction makes it a proper function.
  • While many angles have a sine of 0.5, arcsin(0.5) returns only π/6.
• In Trigonometric Equations: is used to solve equations involving the sine function.
  • If sin(θ) = 0.707, then θ = arcsin(0.707) + 2πn or θ = π - arcsin(0.707) + 2πn, for any integer n.

Variants and Related Words

• Inverse sine: A common synonym and alternative name for .
• sin⁻¹: The standard mathematical notation for . The "-1" denotes the inverse function, not a reciprocal.
• Arccosine (arccos): The inverse function of the cosine.
• Arctangent (arctan): The inverse function of the tangent.

Synonyms

• Inverse sine: The direct descriptive synonym for .

Notes

• The input to must be a number between -1 and 1. It is undefined for values outside this range.
• The output is an angle. Ensure your calculator is set to the correct unit (degrees or radians) when computing .

Noun

1. the inverse function of the sine; the angle that has a sine equal to a given number