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受欢迎的三角函数 >

(2(cos(pi/3)+isin(pi/3)))^3

解答

(2 (cos (\frac{π}{3}) + i sin (\frac{π}{3})))^{3}

解答

- 8

求解步骤

(2 (cos (\frac{π}{3}) + i sin (\frac{π}{3})))^{3}

使用以下普通恒等式:

cos (\frac{π}{3}) = \frac{1}{2}

cos (\frac{π}{3})

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

使用以下普通恒等式:

sin (\frac{π}{3}) = \frac{3}{2}

sin (\frac{π}{3})

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

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

化简

(2 (\frac{1}{2} + i \frac{3}{2}))^{3} : - 8

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

乘

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

i \frac{3}{2}

分式相乘:

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

= \frac{3 i}{2}

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

合并分式

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

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

去除括号:

(a) = a

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

乘

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

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

分式相乘:

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

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

约分:

2

= 1 + 3 i

= (1 + 3 i)^{3}

使用完全立方公式:

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

a = 1, b = 3 i

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

化简

1^{3} + 3 \cdot 1^{2} 3 i + 3 \cdot 1 \cdot (3 i)^{2} + (3 i)^{3} : - 8

1^{3} + 3 \cdot 1^{2} 3 i + 3 \cdot 1 \cdot (3 i)^{2} + (3 i)^{3}

使用法则

1^{a} = 1

1^{3} = 1, 1^{2} = 1

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

3 \cdot 1 \cdot 3 i = 33 i

3 \cdot 1 \cdot 3 i

数字相乘:

3 \cdot 1 = 3

= 33 i

3 \cdot 1 \cdot (3 i)^{2} = - 9

3 \cdot 1 \cdot (3 i)^{2}

(3 i)^{2} = 3 i^{2}

(3 i)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{2} (3)^{2}

(3)^{2} : 3

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 3

= 3 i^{2}

= 3 \cdot 1 \cdot 3 i^{2}

数字相乘:

3 \cdot 1 \cdot 3 = 9

= 9 i^{2}

i^{2} = - 1

i^{2}

使用虚数运算法则:

i^{2} = - 1

= - 1

= 9 (- 1)

去除括号:

(- a) = - a

= - 9 \cdot 1

数字相乘:

9 \cdot 1 = 9

= - 9

= 1 + 33 i - 9 + (3 i)^{3}

(3 i)^{3} = - 33 i

(3 i)^{3}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= i^{3} (3)^{3}

(3)^{3} : 3^{\frac{3}{2}}

使用根式运算法则:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{3}

使用指数法则:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 3}

\frac{1}{2} \cdot 3 = \frac{3}{2}

\frac{1}{2} \cdot 3

分式相乘:

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

= \frac{1 \cdot 3}{2}

数字相乘:

1 \cdot 3 = 3

= \frac{3}{2}

= 3^{\frac{3}{2}}

= 3^{\frac{3}{2}} i^{3}

3^{\frac{3}{2}} = 33

3^{\frac{3}{2}}

3^{\frac{3}{2}} = 3^{1 + \frac{1}{2}}

= 3^{1 + \frac{1}{2}}

使用指数法则:

x^{a + b} = x^{a} x^{b}

= 3^{1} \cdot 3^{\frac{1}{2}}

整理后得

= 33

= 33 i^{3}

i^{3} = - i

i^{3}

使用指数法则:

a^{b + c} = a^{b} a^{c}

i^{3} = i^{2} i

= i^{2} i

使用虚数运算法则:

i^{2} = - 1

= - 1 i

乘以:

1 i = i

= - i

= 33 (- i)

去除括号:

(- a) = - a

= - 33 i

= 1 + 33 i - 9 - 33 i

对同类项分组

= 33 i - 33 i + 1 - 9

同类项相加:

33 i - 33 i = 0

= 1 - 9

数字相减:

1 - 9 = - 8

= - 8

= - 8

= - 8

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