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受欢迎的三角函数 >

3sinh(x)-cosh(x)=1

解答

3 sinh (x) - cosh (x) = 1

解答

x = ln (2)

+1

度数

x = 39.71440 \dots^{\circ}

求解步骤

3 sinh (x) - cosh (x) = 1

使用三角恒等式改写

3 sinh (x) - cosh (x) = 1

使用双曲函数恒等式:

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

3 \cdot \frac{e ^{x} - e ^{- x}}{2} - cosh (x) = 1

使用双曲函数恒等式:

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

3 \cdot \frac{e ^{x} - e ^{- x}}{2} - \frac{e ^{x} + e ^{- x}}{2} = 1

3 \cdot \frac{e ^{x} - e ^{- x}}{2} - \frac{e ^{x} + e ^{- x}}{2} = 1

3 \cdot \frac{e ^{x} - e ^{- x}}{2} - \frac{e ^{x} + e ^{- x}}{2} = 1 : x = ln (2)

3 \cdot \frac{e ^{x} - e ^{- x}}{2} - \frac{e ^{x} + e ^{- x}}{2} = 1

在两边乘以

2

3 \cdot \frac{e ^{x} - e ^{- x}}{2} \cdot 2 - \frac{e ^{x} + e ^{- x}}{2} \cdot 2 = 1 \cdot 2

化简

3 (e^{x} - e^{- x}) - (e^{x} + e^{- x}) = 2

使用指数运算法则

3 (e^{x} - e^{- x}) - (e^{x} + e^{- x}) = 2

使用指数法则:

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

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

3 (e^{x} - (e^{x})^{- 1}) - (e^{x} + (e^{x})^{- 1}) = 2

3 (e^{x} - (e^{x})^{- 1}) - (e^{x} + (e^{x})^{- 1}) = 2

用

e^{x} = u

改写方程式

3 (u - (u)^{- 1}) - (u + (u)^{- 1}) = 2

解

3 (u - u^{- 1}) - (u + u^{- 1}) = 2 : u = 2, u = - 1

3 (u - u^{- 1}) - (u + u^{- 1}) = 2

整理后得

3 (u - \frac{1}{u}) - (u + \frac{1}{u}) = 2

化简

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

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

打开括号

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

使用加减运算法则

+ (- a) = - a

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

3 (u - \frac{1}{u}) - u - \frac{1}{u} = 2

在两边乘以

u

3 (u - \frac{1}{u}) - u - \frac{1}{u} = 2

在两边乘以

u

3 (u - \frac{1}{u}) u - uu - \frac{1}{u} u = 2 u

化简

3 (u - \frac{1}{u}) u - uu - \frac{1}{u} u = 2 u

化简

- uu : - u^{2}

- uu

使用指数法则:

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

uu = u^{1 + 1}

= - u^{1 + 1}

数字相加:

1 + 1 = 2

= - u^{2}

化简

- \frac{1}{u} u : - 1

- \frac{1}{u} u

分式相乘:

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

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

约分:

u

= - 1

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

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

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

展开

3 (u - \frac{1}{u}) u - u^{2} - 1 : 2 u^{2} - 4

3 (u - \frac{1}{u}) u - u^{2} - 1

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

乘开

3 u (u - \frac{1}{u}) : 3 u^{2} - 3

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

使用分配律:

a (b - c) = ab - a c

a = 3 u, b = u, c = \frac{1}{u}

= 3 uu - 3 u \frac{1}{u}

= 3 uu - 3 \cdot \frac{1}{u} u

化简

3 uu - 3 \cdot \frac{1}{u} u : 3 u^{2} - 3

3 uu - 3 \cdot \frac{1}{u} u

3 uu = 3 u^{2}

3 uu

使用指数法则:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

数字相加:

1 + 1 = 2

= 3 u^{2}

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

3 \cdot \frac{1}{u} u

分式相乘:

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

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

约分:

u

= 1 \cdot 3

数字相乘:

1 \cdot 3 = 3

= 3

= 3 u^{2} - 3

= 3 u^{2} - 3

= 3 u^{2} - 3 - u^{2} - 1

化简

3 u^{2} - 3 - u^{2} - 1 : 2 u^{2} - 4

3 u^{2} - 3 - u^{2} - 1

对同类项分组

= 3 u^{2} - u^{2} - 3 - 1

同类项相加:

3 u^{2} - u^{2} = 2 u^{2}

= 2 u^{2} - 3 - 1

数字相减:

- 3 - 1 = - 4

= 2 u^{2} - 4

= 2 u^{2} - 4

2 u^{2} - 4 = 2 u

解

2 u^{2} - 4 = 2 u : u = 2, u = - 1

2 u^{2} - 4 = 2 u

将

2 u

para o lado esquerdo

2 u^{2} - 4 = 2 u

两边减去

2 u

2 u^{2} - 4 - 2 u = 2 u - 2 u

化简

2 u^{2} - 4 - 2 u = 0

2 u^{2} - 4 - 2 u = 0

改写成标准形式

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

2 u^{2} - 2 u - 4 = 0

使用求根公式求解

2 u^{2} - 2 u - 4 = 0

二次方程求根公式:

若

a = 2, b = - 2, c = - 4

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

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

(- 2)^{2} - 4 \cdot 2 (- 4) = 6

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 2 \cdot 4 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数字相加:

4 + 32 = 36

= 36

因式分解数字:

36 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a

6^{2} = 6

= 6

u_{1, 2} = \frac{- ( - 2 ) \pm 6}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 6}{2 \cdot 2}, u_{2} = \frac{- ( - 2 ) - 6}{2 \cdot 2}

u = \frac{- ( - 2 ) + 6}{2 \cdot 2} : 2

\frac{- ( - 2 ) + 6}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{2 + 6}{2 \cdot 2}

数字相加:

2 + 6 = 8

= \frac{8}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= 2

u = \frac{- ( - 2 ) - 6}{2 \cdot 2} : - 1

\frac{- ( - 2 ) - 6}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{2 - 6}{2 \cdot 2}

数字相减:

2 - 6 = - 4

= \frac{- 4}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{4}

使用分式法则:

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

= - \frac{4}{4}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = 2, u = - 1

u = 2, u = - 1

验证解

找到无定义的点（奇点）:

u = 0

取

3 (u - u^{- 1}) - (u + u^{- 1})

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 = 2, u = - 1

u = 2, u = - 1

代回

u = e^{x}

，求解

x

解

e^{x} = 2 : x = ln (2)

e^{x} = 2

使用指数运算法则

e^{x} = 2

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (2)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (2)

x = ln (2)

解

e^{x} = - 1 : x \in R

无解

e^{x} = - 1

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = ln (2)

x = ln (2)

作图

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