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受欢迎的三角函数 >

sinh(x)= 4/3

解答

sinh (x) = \frac{4}{3}

解答

x = ln (3)

+1

度数

x = 62.94584 \dots^{\circ}

求解步骤

sinh (x) = \frac{4}{3}

使用三角恒等式改写

sinh (x) = \frac{4}{3}

使用双曲函数恒等式:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{4}{3}

\frac{e ^{x} - e ^{- x}}{2} = \frac{4}{3}

\frac{e ^{x} - e ^{- x}}{2} = \frac{4}{3} : x = ln (3)

\frac{e ^{x} - e ^{- x}}{2} = \frac{4}{3}

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 3 = 2 \cdot 4

化简

(e^{x} - e^{- x}) \cdot 3 = 8

使用指数运算法则

(e^{x} - e^{- x}) \cdot 3 = 8

使用指数法则:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 3 = 8

(e^{x} - (e^{x})^{- 1}) \cdot 3 = 8

用

e^{x} = u

改写方程式

(u - (u)^{- 1}) \cdot 3 = 8

解

(u - u^{- 1}) \cdot 3 = 8 : u = 3, u = - \frac{1}{3}

(u - u^{- 1}) \cdot 3 = 8

整理后得

(u - \frac{1}{u}) \cdot 3 = 8

化简

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

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

使用交换律:

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

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

3 (u - \frac{1}{u}) = 8

展开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

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

3 \cdot \frac{1}{u}

分式相乘:

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

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

数字相乘:

1 \cdot 3 = 3

= \frac{3}{u}

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

3 u - \frac{3}{u} = 8

在两边乘以

u

3 u - \frac{3}{u} = 8

在两边乘以

u

3 uu - \frac{3}{u} u = 8 u

化简

3 uu - \frac{3}{u} u = 8 u

化简

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}

化简

- \frac{3}{u} u : - 3

- \frac{3}{u} u

分式相乘:

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

= - \frac{3 u}{u}

约分:

u

= - 3

3 u^{2} - 3 = 8 u

3 u^{2} - 3 = 8 u

3 u^{2} - 3 = 8 u

解

3 u^{2} - 3 = 8 u : u = 3, u = - \frac{1}{3}

3 u^{2} - 3 = 8 u

将

8 u

para o lado esquerdo

3 u^{2} - 3 = 8 u

两边减去

8 u

3 u^{2} - 3 - 8 u = 8 u - 8 u

化简

3 u^{2} - 3 - 8 u = 0

3 u^{2} - 3 - 8 u = 0

改写成标准形式

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

3 u^{2} - 8 u - 3 = 0

使用求根公式求解

3 u^{2} - 8 u - 3 = 0

二次方程求根公式:

若

a = 3, b = - 8, c = - 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 ( - 3 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 3 ( - 3 )}{2 \cdot 3}

(- 8)^{2} - 4 \cdot 3 (- 3) = 10

(- 8)^{2} - 4 \cdot 3 (- 3)

使用法则

- (- a) = a

= (- 8)^{2} + 4 \cdot 3 \cdot 3

使用指数法则:

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

若

n

是偶数

(- 8)^{2} = 8^{2}

= 8^{2} + 4 \cdot 3 \cdot 3

数字相乘:

4 \cdot 3 \cdot 3 = 36

= 8^{2} + 36

8^{2} = 64

= 64 + 36

数字相加:

64 + 36 = 100

= 100

因式分解数字:

100 = 1 0^{2}

= 1 0^{2}

使用根式运算法则:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- ( - 8 ) \pm 10}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 8 ) + 10}{2 \cdot 3}, u_{2} = \frac{- ( - 8 ) - 10}{2 \cdot 3}

u = \frac{- ( - 8 ) + 10}{2 \cdot 3} : 3

\frac{- ( - 8 ) + 10}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 + 10}{2 \cdot 3}

数字相加:

8 + 10 = 18

= \frac{18}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{18}{6}

数字相除:

\frac{18}{6} = 3

= 3

u = \frac{- ( - 8 ) - 10}{2 \cdot 3} : - \frac{1}{3}

\frac{- ( - 8 ) - 10}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 - 10}{2 \cdot 3}

数字相减:

8 - 10 = - 2

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

数字相乘:

2 \cdot 3 = 6

= \frac{- 2}{6}

使用分式法则:

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

= - \frac{2}{6}

约分:

2

= - \frac{1}{3}

二次方程组的解是:

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

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

验证解

找到无定义的点（奇点）:

u = 0

取

(u - u^{- 1}) 3

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

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

代回

u = e^{x}

，求解

x

解

e^{x} = 3 : x = ln (3)

e^{x} = 3

使用指数运算法则

e^{x} = 3

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (3)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3)

x = ln (3)

解

e^{x} = - \frac{1}{3} : x \in R

无解

e^{x} = - \frac{1}{3}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = ln (3)

x = ln (3)

作图

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