Arctangent, denoted as $\arctan(x)$ or $\tan^{-1}(x)$, is the inverse function of the tangent function. It returns the angle whose tangent is a given number.
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The range of $\arctan(x)$ is from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.
$\arctan(1) = \frac{\pi}{4}$ because $\tan(\frac{\pi}{4}) = 1$.
$$y = \arctan(x)$$ if and only if $$x = \tan(y)$$ where $$-\frac{\pi}{2} < y < \frac{\pi}{2}$$.
Arctangent can be used to solve for angles in right triangles when the opposite and adjacent sides are known.
The derivative of $y = \arctan(x)$ with respect to $x$ is $$\frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^2}$$.
Review Questions
What is the range of the arctangent function?
If $y = \arctan(3)$, what is $x$ in terms of tangent?
Calculate the derivative of $y = \arctan(x)$ with respect to $x$.
Related terms
Tangent: A trigonometric function that relates the opposite side to the adjacent side of a right triangle, defined as $\tan(θ) = \frac{opposite}{adjacent}$.
Inverse Trigonometric Functions: Functions that reverse trigonometric functions such as sine, cosine, and tangent to obtain an angle from a ratio.
\texttt{Pi (π)}: A mathematical constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter.