簡約 (cos(x)+isin(x))^7

解

簡約

(cos (x) + i sin (x))^{7}

(cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + sin (x) (42 cos^{6} (x) + 21 sin^{4} (x) - 22 sin^{6} (x) - 35 cos^{4} (x)) i

解答ステップ

(cos (x) + i sin (x))^{7}

拡張

(cos (x) + i sin (x))^{7} : cos^{7} (x) + 7 i cos^{6} (x) sin (x) - 21 cos^{5} (x) sin^{2} (x) - 35 i cos^{4} (x) sin^{3} (x) + 35 sin^{4} (x) cos^{3} (x) + 21 i sin^{5} (x) cos^{2} (x) - 7 sin^{6} (x) cos (x) - i sin^{7} (x)

= cos^{7} (x) + 7 i cos^{6} (x) sin (x) - 21 cos^{5} (x) sin^{2} (x) - 35 i cos^{4} (x) sin^{3} (x) + 35 sin^{4} (x) cos^{3} (x) + 21 i sin^{5} (x) cos^{2} (x) - 7 sin^{6} (x) cos (x) - i sin^{7} (x)

標準的な複素数形式で書き直す:

(cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + (7 cos^{6} (x) sin (x) - 35 cos^{4} (x) sin^{3} (x) + 21 sin^{5} (x) cos^{2} (x) - sin^{7} (x)) i

= (cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + (7 cos^{6} (x) sin (x) - 35 cos^{4} (x) sin^{3} (x) + 21 sin^{5} (x) cos^{2} (x) - sin^{7} (x)) i

簡素化

7 cos^{6} (x) sin (x) - 35 cos^{4} (x) sin^{3} (x) + 21 sin^{5} (x) cos^{2} (x) - sin^{7} (x) : sin (x) (42 cos^{6} (x) + 21 sin^{4} (x) - 22 sin^{6} (x) - 35 cos^{4} (x))

= (cos^{7} (x) - 21 cos^{5} (x) sin^{2} (x) + 35 sin^{4} (x) cos^{3} (x) - 7 sin^{6} (x) cos (x)) + sin (x) (42 cos^{6} (x) + 21 sin^{4} (x) - 22 sin^{6} (x) - 35 cos^{4} (x)) i