Question

定义：e^iθ=cosθ+isinθ,其中i为虚数单位，e为自然对数的底数，θ∈R,且实数指数幂的运算性质对e^iθ都适用.

若a=cos⁵θ-cos³θsin²θ+cosθsin⁴θ,b=cos⁴θsinθ-cos²θsin³θ+sin⁵θ,

则复数a+bi等于

A.cos5θ+isin5θ                    B.cos5θ-isin5θ

C.sin5θ+icos5θ                    D.sin5θ-icos5θ

Answer 1

A?

解析:(cosθ+isinθ)⁵=cos⁵θ+cos⁴θsinθ·i-cos³θ-cos²θsin³θ·i

-cos²θsin³θi+cosθsin⁴θ+sin⁵θ·i=a+bi.?

∵E^iθ=cosθ+isinθ,?

∴a+bi=(E^iθ)⁵=E^i·5θ=cos5θ+isin5θ.

选A.