展開する (xcos(a)+ycos(a+b)+zcos(a+b+c))^2

解

展開する

(x cos (a) + y cos (a + b) + z cos (a + b + c))^{2}

x^{2} cos^{2} (a) + x y cos (a) cos (a + b) + x z cos (a) cos (a + b + c) + x y cos (a + b) cos (a) + y^{2} cos^{2} (a + b) + yz cos (a + b) cos (a + b + c) + x z cos (a + b + c) cos (a) + yz cos (a + b + c) cos (a + b) + z^{2} cos^{2} (a + b + c)

解答ステップ

(x cos (a) + y cos (a + b) + z cos (a + b + c))^{2}

指数の規則を適用する:

a^{b + c} = a^{b} \cdot a^{c}

(x cos (a) + y cos (a + b) + z cos (a + b + c))^{2} = (x cos (a) + y cos (a + b) + z cos (a + b + c)) (x cos (a) + y cos (a + b) + z cos (a + b + c))

= (x cos (a) + y cos (a + b) + z cos (a + b + c)) (x cos (a) + y cos (a + b) + z cos (a + b + c))

括弧を分配する

= x cos (a) x cos (a) + x cos (a) y cos (a + b) + x cos (a) z cos (a + b + c) + y cos (a + b) x cos (a) + y cos (a + b) y cos (a + b) + y cos (a + b) z cos (a + b + c) + z cos (a + b + c) x cos (a) + z cos (a + b + c) y cos (a + b) + z cos (a + b + c) z cos (a + b + c)

簡素化

x cos (a) x cos (a) + x cos (a) y cos (a + b) + x cos (a) z cos (a + b + c) + y cos (a + b) x cos (a) + y cos (a + b) y cos (a + b) + y cos (a + b) z cos (a + b + c) + z cos (a + b + c) x cos (a) + z cos (a + b + c) y cos (a + b) + z cos (a + b + c) z cos (a + b + c) : x^{2} cos^{2} (a) + x y cos (a) cos (a + b) + x z cos (a) cos (a + b + c) + x y cos (a + b) cos (a) + y^{2} cos^{2} (a + b) + yz cos (a + b) cos (a + b + c) + x z cos (a + b + c) cos (a) + yz cos (a + b + c) cos (a + b) + z^{2} cos^{2} (a + b + c)

= x^{2} cos^{2} (a) + x y cos (a) cos (a + b) + x z cos (a) cos (a + b + c) + x y cos (a + b) cos (a) + y^{2} cos^{2} (a + b) + yz cos (a + b) cos (a + b + c) + x z cos (a + b + c) cos (a) + yz cos (a + b + c) cos (a + b) + z^{2} cos^{2} (a + b + c)