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## Seno y Coseno a partir de la Fórmula de Euler

A lo largo de las últimas tres entradas hemos estado recorriendo puntos de vista vinculados a la Fórmula de Euler. Sin embargo, aún nos queda un detalle no menor relacionado a ella.

Una consecuencia simple pero importante de la fórmula de Euler es que las funciones seno y coseno pueden ser construidas a partir de la función exponencial. Más precisamente, una mirada a conciencia de la Figura 1 nos revela que

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

Figura 1

mientras que de la Figura 2 deducimos que

$e^{i\theta}-e^{-i\theta}=2i\sin\theta$

Figura 2

Así, obtenemos las tan conocidas fórmulas del campo complejo:

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

$\sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}$

La perspectiva de ver gráficamente estos hechos nos permite ahorrarnos el sencillo pero un tanto tedioso hecho de deducirlas trabajando unos minutos algebraicamente…

Fuente bibliográfica:

• Tristan Needham (1997) Visual Complex Analysis. Oxford University Press

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