The expression "arctan" is a function in mathematics, specifically the arctangent function. It is the inverse of the tangent function, meaning:
$$ \text{If } y = \tan(x), \text{ then } x = \arctan(y) $$
Definition:
$$ \arctan(x) = \theta \quad \text{if and only if} \quad \tan(\theta) = x $$
Domain and Range:
- Domain: All real numbers (i.e., $ x \in \mathbb{R} $)
- Range: $ \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) $
Example:
$$ \arctan(1) = \frac{\pi}{4} $$ $$ \arctan(0) = 0 $$ $$ \arctan(\infty) = \frac{\pi}{2} $$
Graph:
The graph of $ y = \arctan(x) $ is a curve that passes through the points $ (0, 0) $, $ (1, \frac{\pi}{4}) $, $ (-1, -\frac{\pi}{4}) $, and approaches $ \frac{\pi}{2} $ as $ x \to \infty $, and $ -\frac{\pi}{2} $ as $ x \to -\infty $.
Applications:
- In trigonometry, physics, engineering, and calculus.
- Used to find angles in right triangles, or to solve equations involving tangent.
Common Values:
$$ \begin{align} \arctan(0) &= 0 \ \arctan(1) &= \frac{\pi}{4} \ \arctan(\tan(\theta)) &= \theta \quad \text{for } -\frac{\pi}{2} < \theta < \frac{\pi}{2} \end{align} $$
Let me know if you want to compute a specific value or see how it's used in an equation!