Trigonometría

+ \cos(\alpha - \beta) }{ 2}</math>

$\operatorname {sen} (\alpha )\operatorname {sen} (\beta )={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{2}}$

$\operatorname {sen} (\alpha )\cos(\beta )={\frac {\operatorname {sen} (\alpha +\beta )+\operatorname {sen} (\alpha -\beta )}{2}}$

$\cos(\alpha )\operatorname {sen} (\beta )={\frac {\operatorname {sen} (\alpha +\beta )-\operatorname {sen} (\alpha -\beta )}{2}}$

Ángulo doble

$\operatorname {sen} 2\alpha =2\operatorname {sen} \alpha \cdot \cos \alpha \,\!$

$\cos 2\alpha =\cos ^{2}\alpha -\operatorname {sen} ^{2}\alpha \,\!$

$\cos 2\alpha =1-2\operatorname {sen} ^{2}\alpha \,\!$

$\cos 2\alpha =-1+2\cos ^{2}\alpha \,\!$

$\tan 2\alpha ={\frac {2\tan \alpha }{1-\tan ^{2}\alpha }}$

$\operatorname {sen} ^{2}\alpha ={\frac {1-\cos 2\alpha }{2}}$

$\cos ^{2}\alpha ={\frac {1+\cos 2\alpha }{2}}$

Ángulo mitad

$\operatorname {sen} \left({\frac {\alpha }{2}}\right)={\sqrt {\frac {1-\cos \alpha }{2}}}\,\!$

$\cos \left({\frac {\alpha }{2}}\right)={\sqrt {\frac {1+\cos \alpha }{2}}}\,\!$

$\tan \left({\frac {\alpha }{2}}\right)={\sqrt {\frac {1-\cos \alpha }{1+\cos \alpha }}}$

${\begin{aligned}&{\mathrm {\vdash } \tan {\mathrm {(} }{\frac {u}{2}}{\mathrm {)} }\mathrm {=} {\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{\sin \mathrm {(} u\mathrm {)} }}}\\&{}\\&{{i}{\mathrm {)} }\;{Se}\;{parte}\;{de}\;\tan {\mathrm {(} }{x}{\mathrm {)} }\mathrm {=} {\frac {\sin \mathrm {(} x\mathrm {)} }{\cos \mathrm {(} x\mathrm {)} }}{\mathrm {,} }\;{entonces}{\mathrm {:} }}\\&{}\\&{\tan {\mathrm {(} }{\frac {u}{2}}{\mathrm {)} }\mathrm {=} {\frac {\sqrt {\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{2}}}{\sqrt {\frac {{1}\mathrm {+} \cos {\mathrm {(} }{u}{\mathrm {)} }}{2}}}}}\\&{}\\&{{ii}{\mathrm {)} }\;{Se}\;{multiplica}\;{por}\;{uno}\;{para}\;{\mathrm {eliminar} }\;{las}\;{raices}\;{de}\;{dos}{\mathrm {:} }}\\&{}\\&{\;\;\;\;\;\;\;\;\;\;\;\;\mathrm {=} {\frac {\sqrt {2}}{\sqrt {2}}}\mathrm {\cdot } {\frac {\mathrm {(} {\tfrac {\sqrt {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}}{\sqrt {2}}}\mathrm {)} }{\mathrm {(} {\tfrac {\sqrt {{1}\mathrm {+} \cos {\mathrm {(} }{u}{\mathrm {)} }}}{\sqrt {2}}}\mathrm {)} }}}\\&{}\\&{\;\;\;\;\;\;\;\;\;\;\;\;\mathrm {=} {\frac {\sqrt {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}}{\sqrt {{1}\mathrm {+} \cos {\mathrm {(} }{u}{\mathrm {)} }}}}}\\&{}\\&{{iii}{\mathrm {)} }\;{Se}\;{multiplica}\;{otra}\;{vez}\;{por}\;{un}\;{uno}\;{para}\;{obtener}{\mathrm {:} }}\\&{}\\&{\tan {\mathrm {(} }{\frac {u}{2}}{\mathrm {)} }\mathrm {=} {\sqrt {\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{{1}\mathrm {+} \cos {\mathrm {(} }{u}{\mathrm {)} }}}}\mathrm {\cdot } {\sqrt {\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}}}}\\&{}\\&{\;\;\;\;\;\;\;\;\;\;\;\;\mathrm {=} {\sqrt {{\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{{1}\mathrm {+} \cos {\mathrm {(} }{u}{\mathrm {)} }}}\mathrm {\cdot } {\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}}}}}\\&{}\\&{\;\;\;\;\;\;\;\;\;\;\;\;\mathrm {=} {\sqrt {\frac {{\mathrm {(} }{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {))} }^{2}}{{1}\mathrm {-} {\cos }^{2}{\mathrm {(} }{u}{\mathrm {)} }}}}}\\&{}\\&{{iv}{\mathrm {)} }\;{Utilizando}\;{sen}^{2}{x}\mathrm {=} {1}\mathrm {-} {\cos }^{2}{x}{\mathrm {,} }\;{se}\;{obtiene}{\mathrm {:} }}\\&{}\\&{\tan {\mathrm {(} }{\frac {u}{2}}{\mathrm {)} }\mathrm {=} {\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{\sqrt {{\sin }^{2}\mathrm {(} u\mathrm {)} }}}}\\&{}\\&{\;\;\;\;\;\;\;\;\;\;\;\;\mathrm {=} {\frac {{1}\mathrm {-} \cos {\mathrm {(} }{u}{\mathrm {)} }}{\sin \mathrm {(} u\mathrm {)} }}}\\&{}\\&{\mathrm {\therefore } \;{La}\;{igualdad}\;{queda}\;{demostrada}{\mathrm {.} }}\end{aligned}}$

Otras identidades trigonométricas

$\operatorname {sen} \left({\frac {\pi }{2}}-\alpha \right)=\cos \alpha \,\!$

$\cos \left({\frac {\pi }{2}}-\alpha \right)=\operatorname {sen} \alpha \,\!$

$\operatorname {sen} (\pi -\alpha )=\operatorname {sen} \alpha \,\!$

$\cos(\pi -\alpha )=-\cos \alpha \,\!$

$\operatorname {sen} (2\pi -\alpha )=-\operatorname {sen} \alpha \,\!$

$\cos(2\pi -\alpha )=\cos \alpha \,\!$

$\operatorname {sen} \alpha \cdot \cos \alpha +\operatorname {sen} \beta \cdot \cos \beta =\operatorname {sen} (\alpha +\beta )\cdot \cos(\alpha -\beta )$

Error al representar (Error de conversión. El servidor ("https://wikimedia.org/api/rest_") informó: "Cannot get mml. 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}}{\mathrm {)} }}\\&{}\\&{{ii}{\mathrm {.} }{i}{\mathrm {)} }{Se}\;{opera}\;{con}\;{el}\;\sin {\mathrm {(} }{x}\mathrm {+} {y}{\mathrm {),} }\;{entonces}{\mathrm {:} }}\\&{}\\&{\sin {\mathrm {(} }{x}\mathrm {+} {y}{\mathrm {)} }\mathrm {=} \sin {\mathrm {(} }{x}{\mathrm {)} }\mathrm {\cdot } \cos {\mathrm {(} }{y}{\mathrm {)} }\mathrm {+} \cos {\mathrm {(} }{x}{\mathrm {)} }\mathrm {\cdot } \sin {\mathrm {(} }{y}{\mathrm {)} }}\\&{}\\&{{iii}{\mathrm {.} }{i}{\mathrm {)} }{Se}\;{obtiene}\;{el}\;{equivalente}\;{del}\;\cos {\mathrm {(} }{x}{\mathrm {):} }}\\&{}\\&{{\cos }^{2}{\mathrm {(} }{x}{\mathrm {)} }\mathrm {+} {\sin }^{2}{\mathrm {(} }{x}{\mathrm {)} }\mathrm {=} {1}}\\&{}\\&{{\cos }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))} }\mathrm {=} {1}\mathrm {-} {\sin }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))} }}\\&{}\\&{{\cos }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))} }\mathrm {=} {1}\mathrm {-} {\mathrm {[} }\sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))} }\mathrm {\cdot } \sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))]} }}\\&{}\\&{{\cos }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))} }\mathrm {=} {1}\mathrm {-} {\mathrm {(} }{\mathit {\alpha }}\mathrm {\cdot } {\mathit {\alpha }}{\mathrm {)} }}\\&{}\\&{\cos {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))} }\mathrm {=} {\sqrt {{1}\mathrm {-} {\mathit {\alpha }}^{2}}}}\\&{}\\&{{iii}{\mathrm {.} }{ii}{\mathrm {)} }\;{Se}\;{obtiene}\;{el}\;{equivalente}\;{de}\;\cos {\mathrm {(} }{y}{\mathrm {)} }\;{\mathrm {:} }}\\&{}\\&{{\cos }^{2}{\mathrm {(} }{y}{\mathrm {)} }\mathrm {+} {\sin }^{2}{\mathrm {(} }{y}{\mathrm {)} }\mathrm {=} {1}}\\&{}\\&{{\cos }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))} }\mathrm {=} {1}\mathrm {-} {\sin }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))} }}\\&{}\\&{{\cos }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))} }\mathrm {=} {1}\mathrm {-} {\mathrm {[} }\sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))} }\mathrm {\cdot } \sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))]} }}\\&{}\\&{{\cos }^{2}{\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))} }\mathrm {=} {1}\mathrm {-} {\mathrm {(} }{\mathit {\beta }}\mathrm {\cdot } {\mathit {\beta }}{\mathrm {)} }}\\&{}\\&{\cos {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))} }\mathrm {=} {\sqrt {{1}\mathrm {-} {\mathit {\beta }}^{2}}}}\\&{}\\&{{ii}{\mathrm {.} }{ii}{\mathrm {)} }{Se}\;{sustituyen}\;{del}\;{paso}\;{iii}{\mathrm {)} }\;{el}\;\cos {\mathrm {(} }{x}{\mathrm {)} }\;{y}\;{el}\;\cos \;{\mathrm {(} }{y}{\mathrm {):} }}\\&{}\\&{\sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {)} }\mathrm {+} \arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))} }\mathrm {=} {\mathrm {\{[} }\sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {))]} }\mathrm {\cdot } {\mathrm {(} }{\sqrt {{1}\mathrm {-} {\mathit {\beta }}^{2}}}{\mathrm {)\}} }\mathrm {+} {\mathrm {\{(} }{\sqrt {{1}\mathrm {-} {\mathit {\alpha }}^{2}}}{\mathrm {)} }\mathrm {\cdot } {\mathrm {[} }\sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))]\}} }}\\&{}\\&{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathrm {=} {\mathrm {[} }{\mathit {\alpha }}\mathrm {\cdot } {\sqrt {{1}\mathrm {-} {\mathit {\beta }}^{2}}}{\mathrm {]} }\mathrm {+} {\mathrm {[} }{\mathit {\beta }}\mathrm {\cdot } {\sqrt {{1}\mathrm {-} {\mathit {\alpha }}^{2}}}{\mathrm {]} }}\\&{}\\&{{iv}{\mathrm {)} }{Se}\;{obtiene}\;{el}\;\arcsin {\mathrm {[} }\sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {)} }\mathrm {+} \arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))],} }\;{para}\;{te}{\mathrm {rmin} }{ar}\;{la}\;{demostracion}{\mathrm {:} }}\\&{}\\&{\arcsin {\mathrm {[} }\sin {\mathrm {(} }\arcsin {\mathrm {(} }{\mathit {\alpha }}{\mathrm {)} }\mathrm {+} \arcsin {\mathrm {(} }{\mathit {\beta }}{\mathrm {))]} }\mathrm {=} \;\arcsin {\mathrm {[} }{\mathit {\alpha }}\mathrm {\cdot } {\sqrt {{1}\mathrm {-} {\mathit {\beta }}^{2}}}\mathrm {+} {\mathit {\beta }}\mathrm {\cdot } {\sqrt {{1}\mathrm {-} {\mathit {\alpha }}^{2}}}{\mathrm {]} }}\\&{}\\&{\mathrm {\therefore } \;{queda}\;{demostrada}\;{la}\;{igualdad}{\mathrm {.} }}\end{aligned}}}

Véase también: Sinusoide

Seno y coseno, funciones complejas

El seno y coseno se definen en matemática compleja, gracias a la fórmula de Euler como:

$\operatorname {sen} \alpha ={\frac {e^{i\alpha }-e^{-i\alpha }}{2i}}$

$\cos \alpha ={\frac {e^{i\alpha }+e^{-i\alpha }}{2}}$

Por lo tanto, la tangente quedará definida como:

$\tan \alpha ={\frac {1}{i}}{\frac {e^{i\alpha }-e^{-i\alpha }}{e^{i\alpha }+e^{-i\alpha }}}=\ {-i}{\frac {e^{i\alpha }-e^{-i\alpha }}{e^{i\alpha }+e^{-i\alpha }}}$

Siendo $i={\sqrt {-1}}$ (también puede representarse como j).

Es preciso destacar, que todas las formulas trigonometricas anteriores, son derivadas del Teorema de Pitágoras.

Véase también

• Historia de la trigonometría
• Función trigonométrica
• Identidad trigonométrica
• Funciones hiperbólicas
• Lista de integrales de funciones trigonométricas
• Fórmula de Euler y Número complejo, para funciones trigonométricas complejas
• Trigonometría esférica

Bibliografía

• Cortés Espinosa de los Monteros, Nuria. Ediciones Didácticas y Pedagógicas S. L., ed. Actividades para unidad didáctica sobre trigonometría [Recurso electrónico] (2008). ISBN 978-84-936336-3-9. 
• Domínguez Muro, Mariano. Universidad de Salamanca. Ediciones Universidad Salamanca, ed. Trigonometría activa: 2 BUP (1985). ISBN 978-84-7800-056-2. 

Enlaces externos

• Wikilibros alberga un libro o manual sobre Tabla trigonométrica.
• Ejercicios de Trigonometría (Proyecto Descartes para Educación Secundaria del Ministerio de Educación de España).
• Álgebra y Trigonometría. Universidad de Chile
• Trigonometría (Applets con Geogebra de Manuel Sada).
• Trigonometría. Precálculo21
• Orígenes de la trigonometría (Webquest).
• Matemática - Trigonometría (Apuntes y ejercicios de Trigonometría en Fisicanet).
• Unidad Didáctica. Razones Trigonométricas (IES López de Arenas, de Marchena (Sevilla)).
• La trigonometría, ¿para qué sirve?
• Funciones trigonométricas (Proyecto Descartes para Educación Secundaria del Ministerio de Educación de España).