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人気のある代数 >

展開する (cos(x)+isin(x))^4

解

展開する

(cos (x) + i sin (x))^{4}

解

(cos^{4} (x) - 6 cos^{2} (x) sin^{2} (x) + sin^{4} (x)) + (4 cos^{3} (x) sin (x) - 4 sin^{3} (x) cos (x)) i

解答ステップ

(cos (x) + i sin (x))^{4}

2項定理を適用する:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = cos (x), b = i sin (x)

= i = 0 \sum 4 (i 4) cos^{(4 - i)} (x) (i sin (x))^{i}

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} cos^{4} (x) (i sin (x))^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} cos^{3} (x) (i sin (x))^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} cos^{2} (x) (i sin (x))^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} cos^{1} (x) (i sin (x))^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} cos^{0} (x) (i sin (x))^{4}

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} cos^{4} (x) (i sin (x))^{0} : cos^{4} (x)

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} cos^{3} (x) (i sin (x))^{1} : 4 i cos^{3} (x) sin (x)

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} cos^{2} (x) (i sin (x))^{2} : 6 i^{2} cos^{2} (x) sin^{2} (x)

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} cos^{1} (x) (i sin (x))^{3} : 4 i^{3} sin^{3} (x) cos (x)

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} cos^{0} (x) (i sin (x))^{4} : i^{4} sin^{4} (x)

= cos^{4} (x) + 4 i cos^{3} (x) sin (x) + 6 i^{2} cos^{2} (x) sin^{2} (x) + 4 i^{3} sin^{3} (x) cos (x) + i^{4} sin^{4} (x)

6 i^{2} cos^{2} (x) sin^{2} (x) = - 6 cos^{2} (x) sin^{2} (x)

4 i^{3} sin^{3} (x) cos (x) = - 4 i sin^{3} (x) cos (x)

i^{4} sin^{4} (x) = sin^{4} (x)

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

標準的な複素数形式で書き直す:

(cos^{4} (x) - 6 cos^{2} (x) sin^{2} (x) + sin^{4} (x)) + (4 cos^{3} (x) sin (x) - 4 sin^{3} (x) cos (x)) i

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

人気の例

(\partial)/(\partial x)((z)(x^2+xy^2z))

\frac{\partial}{\partial x} ((z) (x^{2} + x y^{2} z))

d/(dy)((x)((1+2x)^6))

\frac{d}{d y} ((x) ((1 + 2 x)^{6}))

簡約 3(1-2sin^2(x)cos^2(x))

s im pl i f y 3 (1 - 2 sin^{2} (x) cos^{2} (x))

展開する a(-3,-2)b(5,3)

e x p an d a (- 3, - 2) b (5, 3)

簡約 (sqrt(x+15)+sqrt(x))^2

s im pl i f y (x + 15 + x)^{2}