arc cosine
Definition
- Noun:
- The inverse function of the cosine: The
arc cosineis a mathematical function that, for a given number (representing the cosine of an angle), returns the corresponding angle. Specifically, it returns the angle whose cosine is the given number. The result is typically expressed in radians or degrees.
Usage
- The function is used to find an angle when the value of its cosine is known.
- It is commonly written as or .
- The input value (x) for the must be between -1 and 1, inclusive. The output angle is usually in the range from 0 to π radians (0° to 180°).
Examples
- Noun:
- If cos(θ) = 0.5, then θ = arccos(0.5), which is π/3 radians or 60°.
- To solve for the angle in the triangle, you need to use the
arc cosineof the ratio of the adjacent side to the hypotenuse. - The value
arccos(0)is π/2.
Advanced Usage
- Principal Value: The function typically refers to its principal value, which is the unique angle in the range [0, π] for a given input.
- Multivalued Nature: In a broader complex analysis context, the inverse cosine is a multivalued function. The denotes one of its specific branches (usually the principal branch).
Variants and Related Words
- arccos: A common abbreviation for .
- Calculate
arccos(0.7071)to find the angle.
- Inverse Cosine: A fully synonymous term for .
- cos⁻¹: A standard notation for in mathematical expressions.
Synonyms
- Inverse Cosine: The direct synonym for .
- Anticosine: A very rare, synonymous term.
Related Mathematical Functions
- Arc Sine (arcsin): The inverse function of the sine.
- Arc Tangent (arctan): The inverse function of the tangent.
- Inverse Trigonometric Function: The general category to which belongs.
Noun
-
the inverse function of the cosine; the angle that has a cosine equal to a given number
