A.9 Funciones trigonométricas inversas
- Page ID
- 118229
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Es posible que algunos de ustedes no hayan estudiado funciones trigonométricas inversas en secundaria, sin embargo aún esperamos que las conozca al final del curso.
\[ \arcsin x \nonumber \]
Dominio:\(-1 \leq x \leq 1\)
Rango:\(-\frac{\pi}{2} \leq \arcsin x \leq \frac{\pi}{2}\)
\[ \arccos x \nonumber \]
Dominio:\(-1 \leq x \leq 1\)
Rango:\(0 \leq \arccos x \leq \pi\)
\[ \arctan x \nonumber \]
Dominio: todos los números reales
Rango:\(-\frac{\pi}{2} \lt \arctan x \lt \frac{\pi}{2}\)
Como estas funciones son inversas unas de otras tenemos
\ begin {align*}\ arcsin (\ sin\ theta) &=\ theta & -\ frac {\ pi} {2}\ leq\ theta\ leq\ frac {\ pi} {2}\\\ arccos (\ cos\ theta) &=\ theta & 0\ leq\ theta\ leq\ leq\ pi\\ arctan (\ tan\ theta) &=\ theta & -\ frac {\ pi} {2}\ leq\ theta\ leq\ frac {\ pi} {2}\ end {align*}
y también
\ begin {alinear*}\ sin (\ arcsin x) &= x & -1\ leq x\ leq 1\\\ cos (\ arccos x) &= x & -1\ leq x\ leq 1\\\ tan (\ arctan x) &= x &\ texto {cualquier real} x\ end {alinear*}
\[ \arccsc x \nonumber \]
Dominio:\(|x|\ge 1\)
Rango:\(-\frac{\pi}{2} \leq \arccsc x \leq \frac{\pi}{2}\)
\[ \arccsc x \ne 0 \nonumber \]
\[ \arcsec x \nonumber \]
Dominio:\(|x|\ge 1\)
Rango:\(0 \leq \arcsec x \leq \pi\)
\[ \arcsec x \ne \frac{\pi}{2} \nonumber \]
\[ \arccot x \nonumber \]
Dominio: todos los números reales
Rango:\(0 \lt \arccot x \lt \pi\)
Nuevamente
\ begin {alinear*}\ arccsc (\ csc\ theta) &=\ theta & -\ frac {\ pi} {2}\ leq\ theta\ leq\ frac {\ pi} {2},\\ theta\ ne 0\\\ arcsec (\ sec\ theta) & =\ theta & 0\ leq\ theta\ leq\ pi,\\ theta\ ne\ frac {\ pi} {2}\\\ arccot (\ cot\ theta) & =\ theta & 0\ lt\ theta\ lt\ pi\ end {alinear*}
y
\ begin {align*}\ csc (\ arccsc x) &= x & |x|\ ge 1\\\ sec (\ arcsec x) &= x & |x|\ ge 1\\\ cot (\ arccot x) &= x &\ texto {real} x\ end {align*}