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受欢迎的三角函数 >

sinh(x)= 33/56

解答

sinh (x) = \frac{33}{56}

解答

x = ln (\frac{7}{4})

+1

度数

x = 32.06362 \dots^{\circ}

求解步骤

sinh (x) = \frac{33}{56}

使用三角恒等式改写

sinh (x) = \frac{33}{56}

使用双曲函数恒等式:

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56}

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56}

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56} : x = ln (\frac{7}{4})

\frac{e ^{x} - e ^{- x}}{2} = \frac{33}{56}

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 56 = 2 \cdot 33

化简

(e^{x} - e^{- x}) \cdot 56 = 66

使用指数运算法则

(e^{x} - e^{- x}) \cdot 56 = 66

使用指数法则:

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

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

(e^{x} - (e^{x})^{- 1}) \cdot 56 = 66

(e^{x} - (e^{x})^{- 1}) \cdot 56 = 66

用

e^{x} = u

改写方程式

(u - (u)^{- 1}) \cdot 56 = 66

解

(u - u^{- 1}) \cdot 56 = 66 : u = \frac{7}{4}, u = - \frac{4}{7}

(u - u^{- 1}) \cdot 56 = 66

整理后得

(u - \frac{1}{u}) \cdot 56 = 66

化简

(u - \frac{1}{u}) \cdot 56 : 56 (u - \frac{1}{u})

(u - \frac{1}{u}) \cdot 56

使用交换律:

(u - \frac{1}{u}) \cdot 56 = 56 (u - \frac{1}{u})

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

56 (u - \frac{1}{u}) = 66

展开

56 (u - \frac{1}{u}) : 56 u - \frac{56}{u}

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

使用分配律:

a (b - c) = ab - a c

a = 56, b = u, c = \frac{1}{u}

= 56 u - 56 \cdot \frac{1}{u}

56 \cdot \frac{1}{u} = \frac{56}{u}

56 \cdot \frac{1}{u}

分式相乘:

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

= \frac{1 \cdot 56}{u}

数字相乘:

1 \cdot 56 = 56

= \frac{56}{u}

= 56 u - \frac{56}{u}

56 u - \frac{56}{u} = 66

在两边乘以

u

56 u - \frac{56}{u} = 66

在两边乘以

u

56 uu - \frac{56}{u} u = 66 u

化简

56 uu - \frac{56}{u} u = 66 u

化简

56 uu : 56 u^{2}

56 uu

使用指数法则:

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

uu = u^{1 + 1}

= 56 u^{1 + 1}

数字相加:

1 + 1 = 2

= 56 u^{2}

化简

- \frac{56}{u} u : - 56

- \frac{56}{u} u

分式相乘:

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

= - \frac{56 u}{u}

约分:

u

= - 56

56 u^{2} - 56 = 66 u

56 u^{2} - 56 = 66 u

56 u^{2} - 56 = 66 u

解

56 u^{2} - 56 = 66 u : u = \frac{7}{4}, u = - \frac{4}{7}

56 u^{2} - 56 = 66 u

将

66 u

para o lado esquerdo

56 u^{2} - 56 = 66 u

两边减去

66 u

56 u^{2} - 56 - 66 u = 66 u - 66 u

化简

56 u^{2} - 56 - 66 u = 0

56 u^{2} - 56 - 66 u = 0

改写成标准形式

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

56 u^{2} - 66 u - 56 = 0

使用求根公式求解

56 u^{2} - 66 u - 56 = 0

二次方程求根公式:

若

a = 56, b = - 66, c = - 56

u_{1, 2} = \frac{- ( - 66 ) \pm ( - 66 ) ^{2} - 4 \cdot 56 ( - 56 )}{2 \cdot 56}

u_{1, 2} = \frac{- ( - 66 ) \pm ( - 66 ) ^{2} - 4 \cdot 56 ( - 56 )}{2 \cdot 56}

(- 66)^{2} - 4 \cdot 56 (- 56) = 130

(- 66)^{2} - 4 \cdot 56 (- 56)

使用法则

- (- a) = a

= (- 66)^{2} + 4 \cdot 56 \cdot 56

使用指数法则:

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

若

n

是偶数

(- 66)^{2} = 6 6^{2}

= 6 6^{2} + 4 \cdot 56 \cdot 56

数字相乘:

4 \cdot 56 \cdot 56 = 12544

= 6 6^{2} + 12544

6 6^{2} = 4356

= 4356 + 12544

数字相加:

4356 + 12544 = 16900

= 16900

因式分解数字:

16900 = 13 0^{2}

= 13 0^{2}

使用根式运算法则:

n a^{n} = a

13 0^{2} = 130

= 130

u_{1, 2} = \frac{- ( - 66 ) \pm 130}{2 \cdot 56}

将解分隔开

u_{1} = \frac{- ( - 66 ) + 130}{2 \cdot 56}, u_{2} = \frac{- ( - 66 ) - 130}{2 \cdot 56}

u = \frac{- ( - 66 ) + 130}{2 \cdot 56} : \frac{7}{4}

\frac{- ( - 66 ) + 130}{2 \cdot 56}

使用法则

- (- a) = a

= \frac{66 + 130}{2 \cdot 56}

数字相加:

66 + 130 = 196

= \frac{196}{2 \cdot 56}

数字相乘:

2 \cdot 56 = 112

= \frac{196}{112}

约分:

28

= \frac{7}{4}

u = \frac{- ( - 66 ) - 130}{2 \cdot 56} : - \frac{4}{7}

\frac{- ( - 66 ) - 130}{2 \cdot 56}

使用法则

- (- a) = a

= \frac{66 - 130}{2 \cdot 56}

数字相减:

66 - 130 = - 64

= \frac{- 64}{2 \cdot 56}

数字相乘:

2 \cdot 56 = 112

= \frac{- 64}{112}

使用分式法则:

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

= - \frac{64}{112}

约分:

16

= - \frac{4}{7}

二次方程组的解是:

u = \frac{7}{4}, u = - \frac{4}{7}

u = \frac{7}{4}, u = - \frac{4}{7}

验证解

找到无定义的点（奇点）:

u = 0

取

(u - u^{- 1}) 56

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{7}{4}, u = - \frac{4}{7}

u = \frac{7}{4}, u = - \frac{4}{7}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{7}{4} : x = ln (\frac{7}{4})

e^{x} = \frac{7}{4}

使用指数运算法则

e^{x} = \frac{7}{4}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{7}{4})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{7}{4})

x = ln (\frac{7}{4})

解

e^{x} = - \frac{4}{7} : x \in R

无解

e^{x} = - \frac{4}{7}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = ln (\frac{7}{4})

x = ln (\frac{7}{4})

作图

流行的例子

2sin(x)=sqrt(cos(2x)+2)

2 sin (x) = cos (2 x) + 2

tan (x) = \frac{7}{12}

cos(x)=sqrt(1-sin(x))

cos (x) = 1 - sin (x)

-sin(a)-1=3sin(a)+2

- sin (a) - 1 = 3 sin (a) + 2

tan (x) = 7