arctan
Học thuậtThân thiện
Definition
- Noun:
- The inverse function of the tangent:
arctanis a trigonometric function that, for a given number (representing a tangent value), returns the corresponding angle. Specifically, it yields the angle whose tangent is the given number. The result is typically an angle measured in radians.
Usage
- is used to find an angle when the value of the tangent of that angle is known.
- It is often denoted as or , where is a real number.
- The output (principal value) of the function is an angle in the range radians or .
Examples
- Noun/Function:
- If tan(θ) = 1, then θ = arctan(1) = π/4 radians (or 45°).
- To find the angle of elevation, calculate the arctan of the height divided by the distance.
- The value
arctan(0)is 0.
Advanced Usage
- Domain and Range: The function accepts any real number as input (its domain is all real numbers). Its output (range) is restricted to the interval to make it a proper function (single-valued).
- Relation to Complex Numbers: While primarily defined for real arguments, can be extended to handle complex numbers, though this is an advanced mathematical concept.
- In Calculus: The derivative of is , and its integral is .
Variants and Related Words
- arctangent (n): A full, synonymous name for .
- inverse tangent (n): A descriptive synonym for .
- tan⁻¹ (n): An alternative notation for , using exponent-like notation for the inverse function.
- atan (n): A common abbreviation used in computing and programming languages.
Synonyms
- inverse tangent: The descriptive term for the function.
- arctangent: The full name of the function.
Related Mathematical Functions
- arcsin: The inverse function of the sine.
- arccos: The inverse function of the cosine.
- arctan2 (or ): A common two-argument function in computing that computes the arctangent of , taking into account the quadrant of the point to return an angle in the full range .
Noun
- the inverse function of the tangent; the angle that has a tangent equal to a given number