Funciones Trigonométricas Inversas

Notaciones y Definiciones

Notaciones equivalentes:

$$\sin^{-1} x = \arcsin x = \text{áng} \sin x$$ $$\cos^{-1} x = \arccos x = \text{áng} \cos x$$ $$\tan^{-1} x = \arctan x = \text{áng} \tan x$$ $$\cot^{-1} x = \text{arccot} x = \text{áng} \cot x$$

Definición fundamental:

Función	Equivalente a	Dominio	Rango (Valor Principal)
$y = \arcsin x$	$x = \sin y$	$-1 \leq x \leq 1$	$-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$
$y = \arccos x$	$x = \cos y$	$-1 \leq x \leq 1$	$0 \leq y \leq \pi$
$y = \arctan x$	$x = \tan y$	$-\infty < x < \infty$	$-\frac{\pi}{2} < y < \frac{\pi}{2}$
$y = \text{arccot } x$	$x = \cot y$	$-\infty < x < \infty$	$0 < y < \pi$

Relaciones Fundamentales

$$\arccos x = \frac{\pi}{2} - \arcsin x$$ $$\arcctg x = \frac{\pi}{2} - \arctan x$$

Relaciones entre Funciones Inversas (para $x \geq 0$)

Función	Expresiones Equivalentes
$\arcsin x$	$\arccos \sqrt{1-x^2} = \arctan \frac{x}{\sqrt{1-x^2}} = \text{arccot } \frac{\sqrt{1-x^2}}{x}$
$\arccos x$	$\arcsin \sqrt{1-x^2} = \arctan \frac{\sqrt{1-x^2}}{x} = \text{arccot } \frac{x}{\sqrt{1-x^2}}$
$\arctan x$	$\arcsin \frac{x}{\sqrt{1+x^2}} = \arccos \frac{1}{\sqrt{1+x^2}} = \text{arccot } \frac{1}{x}$
$\text{arccot } x$	$\arcsin \frac{1}{\sqrt{1+x^2}} = \arccos \frac{x}{\sqrt{1+x^2}} = \arctan \frac{1}{x}$

Comportamiento con Argumentos Negativos

$$\arcsin(-x) = -\arcsin x$$ $$\arccos(-x) = \pi - \arccos x$$ $$\arctan(-x) = -\arctan x$$ $$\text{arccot}(-x) = \pi - \text{arccot } x$$

Teoremas de Adición

$$\arcsin a \pm \arcsin b = \arcsin\left(a\sqrt{1-b^2} \pm b\sqrt{1-a^2}\right)$$ $$\arccos a \pm \arccos b = \arccos\left(ab \mp \sqrt{(1-a^2)(1-b^2)}\right)$$ $$\arctan a \pm \arctan b = \arctan\left(\frac{a \pm b}{1 \mp ab}\right)$$ $$\arcctg a \pm \arcctg b = \arcctg\left(\frac{ab \mp 1}{b \pm a}\right)$$