高级数学：复数的三角函数式的乘法与除法01

Question

求完整解答步骤

Answer 1

对，复数乘除法，和差的三角函数。

满意，请及时采纳。谢谢！

Answer 2

(1)
z=cosθ +isinθ
(1-z^2)/(1+z^2)
=-1 + 2/(1+z^2)
= -1 + [ 2/(1+cos2θ+isin2θ)]

=-1 + 2(1+cos2θ-isin2θ) /[ (1+cos2θ)^2+(sin2θ)^2 ]
=-1 + 2(1+cos2θ-isin2θ) / (2+2cos2θ)
=-1 + 1 - 2isin2θ/ (2+2cos2θ)
=-isin2θ/ (1+cos2θ)
=-2isinθ.cosθ/ [2(cosθ)^2]
=-itanθ
(2)
z=cosα+isinα, w=cosβ+isinβ
(i)
z/w
=(cosα+isinα)/(cosβ+isinβ)
=(cosα+isinα).(cosβ-isinβ)
=(cosα.cosβ+sinα.sinβ) +i(sinα.cosβ-cosα.sinβ)
=cos(α-β)+isin(α-β)
(ii)
from (i)
z/w + w/z
=cos(α-β)+isin(α-β) + cos(β-α)+isin(β-α)
=cos(α-β)+isin(α-β) + cos(α-β)-isin(α-β)
=2cos(α-β)本回答被网友采纳