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Arctangent

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Algebra and Trigonometry

Definition

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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5 Must Know Facts For Your Next Test

  1. The range of $\arctan(x)$ is from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$.
  2. $\arctan(1) = \frac{\pi}{4}$ because $\tan(\frac{\pi}{4}) = 1$.
  3. $$y = \arctan(x)$$ if and only if $$x = \tan(y)$$ where $$-\frac{\pi}{2} < y < \frac{\pi}{2}$$.
  4. Arctangent can be used to solve for angles in right triangles when the opposite and adjacent sides are known.
  5. 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 $x$.
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