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受欢迎的三角函数 >

sinh(x)= 36/77

解答

sinh (x) = \frac{36}{77}

解答

x = ln (\frac{11}{7})

+1

度数

x = 25.89683 \dots^{\circ}

求解步骤

sinh (x) = \frac{36}{77}

使用三角恒等式改写

sinh (x) = \frac{36}{77}

使用双曲函数恒等式:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77}

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77}

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77} : x = ln (\frac{11}{7})

\frac{e ^{x} - e ^{- x}}{2} = \frac{36}{77}

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 77 = 2 \cdot 36

化简

(e^{x} - e^{- x}) \cdot 77 = 72

使用指数运算法则

(e^{x} - e^{- x}) \cdot 77 = 72

使用指数法则:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 77 = 72

(e^{x} - (e^{x})^{- 1}) \cdot 77 = 72

用

e^{x} = u

改写方程式

(u - (u)^{- 1}) \cdot 77 = 72

解

(u - u^{- 1}) \cdot 77 = 72 : u = \frac{11}{7}, u = - \frac{7}{11}

(u - u^{- 1}) \cdot 77 = 72

整理后得

(u - \frac{1}{u}) \cdot 77 = 72

化简

(u - \frac{1}{u}) \cdot 77 : 77 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 77

使用交换律:

(u - \frac{1}{u}) \cdot 77 = 77 (u - \frac{1}{u})

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

77 (u - \frac{1}{u}) = 72

展开

77 (u - \frac{1}{u}) : 77 u - \frac{77}{u}

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

使用分配律:

a (b - c) = ab - a c

a = 77, b = u, c = \frac{1}{u}

= 77 u - 77 \cdot \frac{1}{u}

77 \cdot \frac{1}{u} = \frac{77}{u}

77 \cdot \frac{1}{u}

分式相乘:

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

= \frac{1 \cdot 77}{u}

数字相乘:

1 \cdot 77 = 77

= \frac{77}{u}

= 77 u - \frac{77}{u}

77 u - \frac{77}{u} = 72

在两边乘以

u

77 u - \frac{77}{u} = 72

在两边乘以

u

77 uu - \frac{77}{u} u = 72 u

化简

77 uu - \frac{77}{u} u = 72 u

化简

77 uu : 77 u^{2}

77 uu

使用指数法则:

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

uu = u^{1 + 1}

= 77 u^{1 + 1}

数字相加:

1 + 1 = 2

= 77 u^{2}

化简

- \frac{77}{u} u : - 77

- \frac{77}{u} u

分式相乘:

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

= - \frac{77 u}{u}

约分:

u

= - 77

77 u^{2} - 77 = 72 u

77 u^{2} - 77 = 72 u

77 u^{2} - 77 = 72 u

解

77 u^{2} - 77 = 72 u : u = \frac{11}{7}, u = - \frac{7}{11}

77 u^{2} - 77 = 72 u

将

72 u

para o lado esquerdo

77 u^{2} - 77 = 72 u

两边减去

72 u

77 u^{2} - 77 - 72 u = 72 u - 72 u

化简

77 u^{2} - 77 - 72 u = 0

77 u^{2} - 77 - 72 u = 0

改写成标准形式

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

77 u^{2} - 72 u - 77 = 0

使用求根公式求解

77 u^{2} - 72 u - 77 = 0

二次方程求根公式:

若

a = 77, b = - 72, c = - 77

u_{1, 2} = \frac{- ( - 72 ) \pm ( - 72 ) ^{2} - 4 \cdot 77 ( - 77 )}{2 \cdot 77}

u_{1, 2} = \frac{- ( - 72 ) \pm ( - 72 ) ^{2} - 4 \cdot 77 ( - 77 )}{2 \cdot 77}

(- 72)^{2} - 4 \cdot 77 (- 77) = 170

(- 72)^{2} - 4 \cdot 77 (- 77)

使用法则

- (- a) = a

= (- 72)^{2} + 4 \cdot 77 \cdot 77

使用指数法则:

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

若

n

是偶数

(- 72)^{2} = 7 2^{2}

= 7 2^{2} + 4 \cdot 77 \cdot 77

数字相乘:

4 \cdot 77 \cdot 77 = 23716

= 7 2^{2} + 23716

7 2^{2} = 5184

= 5184 + 23716

数字相加:

5184 + 23716 = 28900

= 28900

因式分解数字:

28900 = 17 0^{2}

= 17 0^{2}

使用根式运算法则:

n a^{n} = a

17 0^{2} = 170

= 170

u_{1, 2} = \frac{- ( - 72 ) \pm 170}{2 \cdot 77}

将解分隔开

u_{1} = \frac{- ( - 72 ) + 170}{2 \cdot 77}, u_{2} = \frac{- ( - 72 ) - 170}{2 \cdot 77}

u = \frac{- ( - 72 ) + 170}{2 \cdot 77} : \frac{11}{7}

\frac{- ( - 72 ) + 170}{2 \cdot 77}

使用法则

- (- a) = a

= \frac{72 + 170}{2 \cdot 77}

数字相加:

72 + 170 = 242

= \frac{242}{2 \cdot 77}

数字相乘:

2 \cdot 77 = 154

= \frac{242}{154}

约分:

22

= \frac{11}{7}

u = \frac{- ( - 72 ) - 170}{2 \cdot 77} : - \frac{7}{11}

\frac{- ( - 72 ) - 170}{2 \cdot 77}

使用法则

- (- a) = a

= \frac{72 - 170}{2 \cdot 77}

数字相减:

72 - 170 = - 98

= \frac{- 98}{2 \cdot 77}

数字相乘:

2 \cdot 77 = 154

= \frac{- 98}{154}

使用分式法则:

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

= - \frac{98}{154}

约分:

14

= - \frac{7}{11}

二次方程组的解是:

u = \frac{11}{7}, u = - \frac{7}{11}

u = \frac{11}{7}, u = - \frac{7}{11}

验证解

找到无定义的点（奇点）:

u = 0

取

(u - u^{- 1}) 77

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{11}{7}, u = - \frac{7}{11}

u = \frac{11}{7}, u = - \frac{7}{11}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{11}{7} : x = ln (\frac{11}{7})

e^{x} = \frac{11}{7}

使用指数运算法则

e^{x} = \frac{11}{7}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{11}{7})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{11}{7})

x = ln (\frac{11}{7})

解

e^{x} = - \frac{7}{11} : x \in R

无解

e^{x} = - \frac{7}{11}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = ln (\frac{11}{7})

x = ln (\frac{11}{7})

作图

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