arcsine
Noun: - The inverse function of the sine: The arcsine is a mathematical function that, for a given number (the sine value), returns the angle whose sine is that number. It is the principal value of the inverse sine operation.
The arcsine function is used to find an angle when the value of its sine is known. It is commonly applied in trigonometry, geometry, and calculus. The input (the sine value) must be between -1 and 1, inclusive, and the output (the angle) is typically given in radians or degrees, often restricted to a principal range (e.g., -π/2 to π/2 radians or -90° to 90°).
- In a mathematical statement:
- If sin(θ) = 0.5, then θ = arcsine(0.5), which is π/6 radians or 30°.
- In a problem-solving context:
- To find the angle of elevation, calculate the arcsine of the ratio of the opposite side to the hypotenuse.
- Notation: The arcsine function is often denoted as arcsin(x) or sin⁻¹(x). The latter notation can be ambiguous, as it is also used for the reciprocal of the sine function (cosecant) in some contexts, but in modern mathematics, sin⁻¹(x) universally denotes the inverse sine.
- Principal Value: The arcsine function is defined to return a single, principal value. For a real number where -1 ≤ y ≤ 1, arcsin(y) is the unique angle in the range [-π/2, π/2] such that sin(x) = y.
- Inverse sine: A synonym for arcsine.
- Arccosine (noun): The inverse function of the cosine.
- Arctangent (noun): The inverse function of the tangent.
- Inverse sine
- sin⁻¹ (when used as a function symbol)
The arcsine function is specifically the inverse of the sine function. It is not a compound word like "arc length"; the "arc" prefix here refers to the arc of a circle, relating to the angle measurement.
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the inverse function of the sine; the angle that has a sine equal to a given number
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