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yunaarmy43
@yunaarmy43
January 2023
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Soient a et b deux nombres réels
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kixii
Commençons par développer le carré de (a * sin α + b * cos α) :
(a * sin α + b * cos α)² = a² * sin² α + 2ab * sin α * cos α + b² * cos² α
En utilisant l'identité trigonométrique sin² α + cos² α = 1, nous pouvons simplifier cette expression en :
a² * sin² α + 2ab * sin α * cos α + b² * cos² α =
a² * (1 - cos² α) + 2ab * sin α * cos α + b² * cos² α
= a² + b² - a² * cos² α + b² * cos² α
= a² + b²
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(a * sin α + b * cos α)² = a² * sin² α + 2ab * sin α * cos α + b² * cos² α
En utilisant l'identité trigonométrique sin² α + cos² α = 1, nous pouvons simplifier cette expression en :
a² * sin² α + 2ab * sin α * cos α + b² * cos² α =
a² * (1 - cos² α) + 2ab * sin α * cos α + b² * cos² α
= a² + b² - a² * cos² α + b² * cos² α
= a² + b²