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Beliebt Algebra >

erweitern (sin((pix)/L)sin((2piy)/L)-sin((2pix)/L)sin((piy)/L))^2

Lösung

erweitern

(sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) - sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2}

Lösung

sin^{2} (\frac{π x}{L}) sin^{2} (\frac{2 π y}{L}) - 2 sin (\frac{π x}{L}) sin (\frac{π y}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) + sin^{2} (\frac{π y}{L}) sin^{2} (\frac{2 π x}{L})

Schritte zur Lösung

(sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) - sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2}

= (sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) - sin (\frac{π y}{L}) sin (\frac{2 π x}{L}))^{2}

Wende Formel für perfekte quadratische Gleichungen an:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = sin (\frac{π x}{L}) sin (\frac{2 π y}{L}), b = sin (\frac{2 π x}{L}) sin (\frac{π y}{L})

= (sin (\frac{π x}{L}) sin (\frac{2 π y}{L}))^{2} - 2 sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) sin (\frac{π y}{L}) + (sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2}

Vereinfache

(sin (\frac{π x}{L}) sin (\frac{2 π y}{L}))^{2} - 2 sin (\frac{π x}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) sin (\frac{π y}{L}) + (sin (\frac{2 π x}{L}) sin (\frac{π y}{L}))^{2} : sin^{2} (\frac{π x}{L}) sin^{2} (\frac{2 π y}{L}) - 2 sin (\frac{π x}{L}) sin (\frac{π y}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) + sin^{2} (\frac{π y}{L}) sin^{2} (\frac{2 π x}{L})

= sin^{2} (\frac{π x}{L}) sin^{2} (\frac{2 π y}{L}) - 2 sin (\frac{π x}{L}) sin (\frac{π y}{L}) sin (\frac{2 π y}{L}) sin (\frac{2 π x}{L}) + sin^{2} (\frac{π y}{L}) sin^{2} (\frac{2 π x}{L})

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