展开 (xcos(a)+ycos(a+b)+zcos(a+b+c))^2

解答

展开

(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)

求解步骤

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

使用指数法则:

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))

打开括号

= 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)

化简

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)

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e x p an d cos (t) h (t - \frac{π}{2})

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e x p an d (x^{2} + (\frac{2 y}{x})) d x