arc sine
Definition
- Noun:
- The inverse function of the sine: The
arc sineis 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 sineof 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.
Noun
-
the inverse function of the sine; the angle that has a sine equal to a given number
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