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受欢迎的三角函数 >

sinh(x)= 5/12

解答

sinh (x) = \frac{5}{12}

解答

x = ln (\frac{3}{2})

+1

度数

x = 23.23143 \dots^{\circ}

求解步骤

sinh (x) = \frac{5}{12}

使用三角恒等式改写

sinh (x) = \frac{5}{12}

使用双曲函数恒等式:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12}

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12}

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12} : x = ln (\frac{3}{2})

\frac{e ^{x} - e ^{- x}}{2} = \frac{5}{12}

使用分式交叉相乘: 若

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

则

a \cdot d = b \cdot c

(e^{x} - e^{- x}) \cdot 12 = 2 \cdot 5

化简

(e^{x} - e^{- x}) \cdot 12 = 10

使用指数运算法则

(e^{x} - e^{- x}) \cdot 12 = 10

使用指数法则:

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

e^{- x} = (e^{x})^{- 1}

(e^{x} - (e^{x})^{- 1}) \cdot 12 = 10

(e^{x} - (e^{x})^{- 1}) \cdot 12 = 10

用

e^{x} = u

改写方程式

(u - (u)^{- 1}) \cdot 12 = 10

解

(u - u^{- 1}) \cdot 12 = 10 : u = \frac{3}{2}, u = - \frac{2}{3}

(u - u^{- 1}) \cdot 12 = 10

整理后得

(u - \frac{1}{u}) \cdot 12 = 10

化简

(u - \frac{1}{u}) \cdot 12 : 12 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 12

使用交换律:

(u - \frac{1}{u}) \cdot 12 = 12 (u - \frac{1}{u})

12 (u - \frac{1}{u})

12 (u - \frac{1}{u}) = 10

展开

12 (u - \frac{1}{u}) : 12 u - \frac{12}{u}

12 (u - \frac{1}{u})

使用分配律:

a (b - c) = ab - a c

a = 12, b = u, c = \frac{1}{u}

= 12 u - 12 \cdot \frac{1}{u}

12 \cdot \frac{1}{u} = \frac{12}{u}

12 \cdot \frac{1}{u}

分式相乘:

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

= \frac{1 \cdot 12}{u}

数字相乘:

1 \cdot 12 = 12

= \frac{12}{u}

= 12 u - \frac{12}{u}

12 u - \frac{12}{u} = 10

在两边乘以

u

12 u - \frac{12}{u} = 10

在两边乘以

u

12 uu - \frac{12}{u} u = 10 u

化简

12 uu - \frac{12}{u} u = 10 u

化简

12 uu : 12 u^{2}

12 uu

使用指数法则:

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

uu = u^{1 + 1}

= 12 u^{1 + 1}

数字相加:

1 + 1 = 2

= 12 u^{2}

化简

- \frac{12}{u} u : - 12

- \frac{12}{u} u

分式相乘:

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

= - \frac{12 u}{u}

约分:

u

= - 12

12 u^{2} - 12 = 10 u

12 u^{2} - 12 = 10 u

12 u^{2} - 12 = 10 u

解

12 u^{2} - 12 = 10 u : u = \frac{3}{2}, u = - \frac{2}{3}

12 u^{2} - 12 = 10 u

将

10 u

para o lado esquerdo

12 u^{2} - 12 = 10 u

两边减去

10 u

12 u^{2} - 12 - 10 u = 10 u - 10 u

化简

12 u^{2} - 12 - 10 u = 0

12 u^{2} - 12 - 10 u = 0

改写成标准形式

a x^{2} + b x + c = 0

12 u^{2} - 10 u - 12 = 0

使用求根公式求解

12 u^{2} - 10 u - 12 = 0

二次方程求根公式:

若

a = 12, b = - 10, c = - 12

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 12 ( - 12 )}{2 \cdot 12}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 12 ( - 12 )}{2 \cdot 12}

(- 10)^{2} - 4 \cdot 12 (- 12) = 26

(- 10)^{2} - 4 \cdot 12 (- 12)

使用法则

- (- a) = a

= (- 10)^{2} + 4 \cdot 12 \cdot 12

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 10)^{2} = 1 0^{2}

= 1 0^{2} + 4 \cdot 12 \cdot 12

数字相乘:

4 \cdot 12 \cdot 12 = 576

= 1 0^{2} + 576

1 0^{2} = 100

= 100 + 576

数字相加:

100 + 576 = 676

= 676

因式分解数字:

676 = 2 6^{2}

= 2 6^{2}

使用根式运算法则:

n a^{n} = a

2 6^{2} = 26

= 26

u_{1, 2} = \frac{- ( - 10 ) \pm 26}{2 \cdot 12}

将解分隔开

u_{1} = \frac{- ( - 10 ) + 26}{2 \cdot 12}, u_{2} = \frac{- ( - 10 ) - 26}{2 \cdot 12}

u = \frac{- ( - 10 ) + 26}{2 \cdot 12} : \frac{3}{2}

\frac{- ( - 10 ) + 26}{2 \cdot 12}

使用法则

- (- a) = a

= \frac{10 + 26}{2 \cdot 12}

数字相加:

10 + 26 = 36

= \frac{36}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{36}{24}

约分:

12

= \frac{3}{2}

u = \frac{- ( - 10 ) - 26}{2 \cdot 12} : - \frac{2}{3}

\frac{- ( - 10 ) - 26}{2 \cdot 12}

使用法则

- (- a) = a

= \frac{10 - 26}{2 \cdot 12}

数字相减:

10 - 26 = - 16

= \frac{- 16}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{- 16}{24}

使用分式法则:

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

= - \frac{16}{24}

约分:

8

= - \frac{2}{3}

二次方程组的解是:

u = \frac{3}{2}, u = - \frac{2}{3}

u = \frac{3}{2}, u = - \frac{2}{3}

验证解

找到无定义的点（奇点）:

u = 0

取

(u - u^{- 1}) 12

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{3}{2}, u = - \frac{2}{3}

u = \frac{3}{2}, u = - \frac{2}{3}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{3}{2} : x = ln (\frac{3}{2})

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

使用指数运算法则

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

若

f (x) = g (x)

，则

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (\frac{3}{2})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{3}{2})

x = ln (\frac{3}{2})

解

e^{x} = - \frac{2}{3} : x \in R

无解

e^{x} = - \frac{2}{3}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = ln (\frac{3}{2})

x = ln (\frac{3}{2})

作图

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