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受欢迎的三角函数 >

arcsinh(5+12i)

解答

arcsinh (5 + 12 i)

解答

ln (5 + 12 i + - 118 + 120 i)

求解步骤

arcsinh (5 + 12 i)

使用双曲函数恒等式:

arcsinh (x) = ln (x + x^{2} + 1)

= ln (5 + 12 i + (5 + 12 i)^{2} + 1)

化简

ln (5 + 12 i + (5 + 12 i)^{2} + 1) : ln (5 + 12 i + - 118 + 120 i)

ln (5 + 12 i + (5 + 12 i)^{2} + 1)

乘开

(5 + 12 i)^{2} + 1 : - 118 + 120 i

(5 + 12 i)^{2} + 1

(5 + 12 i)^{2} : - 119 + 120 i

使用完全平方公式:

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

a = 5, b = 12 i

= 5^{2} + 2 \cdot 5 \cdot 12 i + (12 i)^{2}

化简

5^{2} + 2 \cdot 5 \cdot 12 i + (12 i)^{2} : - 119 + 120 i

5^{2} + 2 \cdot 5 \cdot 12 i + (12 i)^{2}

5^{2} = 25

5^{2}

5^{2} = 25

= 25

2 \cdot 5 \cdot 12 i = 120 i

2 \cdot 5 \cdot 12 i

数字相乘:

2 \cdot 5 \cdot 12 = 120

= 120 i

(12 i)^{2} = - 144

(12 i)^{2}

使用指数法则:

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

= 1 2^{2} i^{2}

1 2^{2} = 144

= 144 i^{2}

i^{2} = - 1

i^{2}

使用虚数运算法则:

i^{2} = - 1

= - 1

= 144 (- 1)

去除括号:

(- a) = - a

= - 144 \cdot 1

数字相乘:

144 \cdot 1 = 144

= - 144

= 25 + 120 i - 144

数字相减:

25 - 144 = - 119

= - 119 + 120 i

= - 119 + 120 i

= (- 119 + 120 i) + 1

化简

(- 119 + 120 i) + 1 : - 118 + 120 i

(- 119 + 120 i) + 1

去除括号:

(- a) = - a

= - 119 + 120 i + 1

数字相加/相减:

- 119 + 1 = - 118

= - 118 + 120 i

= - 118 + 120 i

= ln (5 + 12 i + - 118 + 120 i)

= ln (5 + 12 i + - 118 + 120 i)

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