arc sine

Thân thiện

The student calculates the arc sine of 0.5 on her calculator.

Definition

1. Noun:
   • The inverse function of the sine: The arc sine is a mathematical function that, for a given number (the sine value), returns the angle whose sine is that number. The result is typically an angle measured in radians or degrees.

Usage

• The function is used to find an angle when the value of its sine is known.
• It is commonly written as arcsin, asin, or sin⁻¹.
• The domain (allowed input values) for the function is from -1 to 1, inclusive. Its principal value range is from -π/2 to π/2 radians (or -90° to 90°).

Examples

• Noun:
  • To find the angle θ, you must calculate the arc sine of 0.5.
  • The value of arc sine(1) is π/2 radians.
  • If sin(θ) = 0.707, then θ = arc sine(0.707), which is approximately 45 degrees.

Advanced Usage

• Principal Value: The function is multivalued. To make it a proper function, its output is restricted to a specific range of angles, known as its principal value branch.
• Relationship with Sine: The undoes the action of the sine function within its restricted range. For a value where -1 ≤ x ≤ 1, sin(arcsin(x)) = x.
• Use in Solving Equations: The is essential for solving trigonometric equations of the form sin(θ) = a.

Variants and Related Words

• arcsin (n): A common abbreviation for .
  • The calculator has a dedicated button for arcsin.
• inverse sine (n): Another name for the function.
  • The inverse sine is one of the basic inverse trigonometric functions.
• Arccosine (n): The inverse function of the cosine.
• Arctangent (n): The inverse function of the tangent.

Synonyms

• Inverse sine: The direct synonym for .

Related Phrases / Concepts

• sin⁻¹: A standard notation for , often used on calculators and in mathematical texts.
  • Be careful not to confuse sin⁻¹(x) with 1/sin(x), which is the cosecant.

Học thuật

Noun

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

Từ đồng nghĩa

• arcsine
• arcsin
• inverse sine