arccosine
Definition
- Noun:
- The inverse function of the cosine: The
arccosineof a number is the angle (usually measured in radians or degrees) whose cosine is that given number. It is the principal value of the inverse cosine function, typically returning an angle in the range from 0 to π radians (0° to 180°).
Usage
- The function, often written as or , is used to find an angle when the value of its cosine is known.
- It is a key function in trigonometry, calculus, and fields requiring angular calculations from ratios, such as physics, engineering, and computer graphics.
Examples
- Noun:
- If cos(θ) = 0.5, then θ = arccos(0.5) = π/3 radians (or 60°).
- To find the angle in the triangle, we calculated the
arccosineof the adjacent side divided by the hypotenuse. - The value of
arccos(-1)is π radians.
Advanced Usage
- Domain and Range: The function is defined for input values (x) in the closed interval [-1, 1]. Its output (the angle) is conventionally restricted to the range [0, π] to make it a proper function (single-valued).
- Mathematical Context: It is one of the principal inverse trigonometric functions, alongside arcsine and arctangent. It solves equations of the form cos(θ) = x for θ.
Variants and Related Words
- Arccos: A common abbreviated notation for .
- Inverse Cosine: A fully descriptive synonym for .
- Arc Cosine: An alternative spelling.
Synonyms
- Inverse Cosine: The direct descriptive synonym.
- cos⁻¹: The standard mathematical notation for the inverse cosine function.
Related Mathematical Concepts
- Arcsine (arcsin): The inverse function of the sine.
- Arctangent (arctan): The inverse function of the tangent.
- Principal Value: The specific angle returned by the inverse trigonometric function within its conventional range.
Noun
-
the inverse function of the cosine; the angle that has a cosine equal to a given number
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