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受欢迎的代数 >

化简 e^{(pi(i-1))/3}

解答

化简

e^{\frac{π ( i - 1 )}{3}}

解答

\frac{1}{2 e ^{\frac{π}{3}}} + i \frac{3}{2 e ^{\frac{π}{3}}}

求解步骤

e^{\frac{π ( i - 1 )}{3}}

使用虚数运算法则:

e^{a + ib} = e^{a} (cos (b) + i sin (b))

= e^{- \frac{π}{3}} (cos (\frac{π}{3}) + i sin (\frac{π}{3}))

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

e^{- \frac{π}{3}} = \frac{1}{e ^{\frac{π}{3}}}

= \frac{1}{e ^{\frac{π}{3}}} (cos (\frac{π}{3}) + i sin (\frac{π}{3}))

cos (\frac{π}{3}) = \frac{1}{2}

sin (\frac{π}{3}) i = \frac{3 i}{2}

= \frac{1}{e ^{\frac{π}{3}}} (\frac{3 i}{2} + \frac{1}{2})

合并分式

\frac{1}{2} + \frac{3 i}{2} : \frac{1 + 3 i}{2}

= (\frac{1 + 3 i}{2}) \frac{1}{e ^{\frac{π}{3}}}

去除括号:

(a) = a

= \frac{1}{e ^{\frac{π}{3}}} \cdot \frac{1 + 3 i}{2}

分式相乘:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot ( 1 + 3 i )}{e ^{\frac{π}{3}} \cdot 2}

1 \cdot (1 + 3 i) = 1 + 3 i

= \frac{1 + 3 i}{2 e ^{\frac{π}{3}}}

将

\frac{1 + 3 i}{e ^{\frac{π}{3}} \cdot 2}

改写成标准复数形式:

\frac{1}{2 e ^{\frac{π}{3}}} + \frac{3}{2 e ^{\frac{π}{3}}} i

= \frac{1}{2 e ^{\frac{π}{3}}} + \frac{3}{2 e ^{\frac{π}{3}}} i

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