化简 (cos(3θ)+isin(3θ))^3

解答

化简

(cos (3 θ) + i sin (3 θ))^{3}

(cos^{3} (3 θ) - 3 sin^{2} (3 θ) cos (3 θ)) + (3 cos^{2} (3 θ) sin (3 θ) - sin^{3} (3 θ)) i

求解步骤

(cos (3 θ) + i sin (3 θ))^{3}

使用完全立方公式:

(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

(cos (3 θ) + i sin (3 θ))^{3} = cos^{3} (3 θ) + 3 cos^{2} (3 θ) i sin (3 θ) + 3 cos (3 θ) (i sin (3 θ))^{2} + (i sin (3 θ))^{3}

= cos^{3} (3 θ) + 3 cos^{2} (3 θ) i sin (3 θ) + 3 cos (3 θ) (i sin (3 θ))^{2} + (i sin (3 θ))^{3}

3 cos^{2} (3 θ) i sin (3 θ) = 3 cos^{2} (3 θ) sin (3 θ) i

3 cos (3 θ) (i sin (3 θ))^{2} = - 3 sin^{2} (3 θ) cos (3 θ)

(i sin (3 θ))^{3} = - sin^{3} (3 θ) i

= cos^{3} (3 θ) + 3 cos^{2} (3 θ) sin (3 θ) i - 3 sin^{2} (3 θ) cos (3 θ) - sin^{3} (3 θ) i

Rewrite in standard complex form:

(cos^{3} (3 θ) - 3 sin^{2} (3 θ) cos (3 θ)) + (3 cos^{2} (3 θ) sin (3 θ) - sin^{3} (3 θ)) i

= (cos^{3} (3 θ) - 3 sin^{2} (3 θ) cos (3 θ)) + (3 cos^{2} (3 θ) sin (3 θ) - sin^{3} (3 θ)) i

e x p an d - 3 (5 - 4 csc (θ) = csc (θ))

e x p an d v a l u e (g - f) (3)

e x p an d lo g_{10} (10000 t)

e x p an d \frac{( x - 1 ) ^{2} ( y - 3 )}{( 3 x - y ) ^{3}}

e x p an d \frac{7 ( x + h ) ^{3} - 3}{( x + h ) ^{4}}