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sinh(x)=(sqrt(2))/2

解答

sinh (x) = \frac{2}{2}

解答

x = ln (\frac{2 + 6}{2})

+1

度数

x = 37.72806 \dots^{\circ}

求解步骤

sinh (x) = \frac{2}{2}

使用三角恒等式改写

sinh (x) = \frac{2}{2}

使用双曲函数恒等式:

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

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

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

\frac{e ^{x} - e ^{- x}}{2} = \frac{2}{2} : x = ln (\frac{2 + 6}{2})

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

使用指数运算法则

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

使用指数法则:

\frac{a ^{m}}{a ^{n}} = a^{m - n}

\frac{2}{2} = 2^{\frac{1}{2} - 1}

\frac{e ^{x} - e ^{- x}}{2} = 2^{\frac{1}{2} - 1}

\frac{1}{2} - 1 = - \frac{1}{2}

\frac{1}{2} - 1

将项转换为分式:

1 = \frac{1 \cdot 2}{2}

= - \frac{1 \cdot 2}{2} + \frac{1}{2}

因为分母相等，所以合并分式:

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

= \frac{- 1 \cdot 2 + 1}{2}

- 1 \cdot 2 + 1 = - 1

- 1 \cdot 2 + 1

数字相乘:

1 \cdot 2 = 2

= - 2 + 1

数字相加/相减:

- 2 + 1 = - 1

= - 1

= \frac{- 1}{2}

使用分式法则:

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

= - \frac{1}{2}

\frac{e ^{x} - e ^{- x}}{2} = 2^{- \frac{1}{2}}

\frac{e ^{x} - e ^{- x}}{2} = 2^{- \frac{1}{2}}

在两边乘以

2

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

化简

2^{- \frac{1}{2}} \cdot 2 : 2

2^{- \frac{1}{2}} \cdot 2

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

2^{- \frac{1}{2}} = \frac{1}{2}

= 2 \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 2}{2}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{2}

使用根式运算法则:

2 = 2^{\frac{1}{2}}

= \frac{2}{2 ^{\frac{1}{2}}}

使用指数法则:

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

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{1 - \frac{1}{2}}

数字相减:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}}

使用根式运算法则:

2^{\frac{1}{2}} = 2

= 2

e^{x} - e^{- x} = 2

使用指数运算法则

e^{x} - e^{- x} = 2

使用指数法则:

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

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

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

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

用

e^{x} = u

改写方程式

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

解

u - u^{- 1} = 2 : u = \frac{2 + 6}{2}, u = \frac{2 - 6}{2}

u - u^{- 1} = 2

整理后得

u - \frac{1}{u} = 2

在两边乘以

u

u - \frac{1}{u} = 2

在两边乘以

u

uu - \frac{1}{u} u = 2 u

化简

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

u^{2} - 1 = 2 u

u^{2} - 1 = 2 u

u^{2} - 1 = 2 u

解

u^{2} - 1 = 2 u : u = \frac{2 + 6}{2}, u = \frac{2 - 6}{2}

u^{2} - 1 = 2 u

将

2 u

para o lado esquerdo

u^{2} - 1 = 2 u

两边减去

2 u

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

化简

u^{2} - 1 - 2 u = 0

u^{2} - 1 - 2 u = 0

改写成标准形式

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

u^{2} - 2 u - 1 = 0

使用求根公式求解

u^{2} - 2 u - 1 = 0

二次方程求根公式:

若

a = 1, b = - 2, c = - 1

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

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

(- 2)^{2} - 4 \cdot 1 \cdot (- 1) = 6

(- 2)^{2} - 4 \cdot 1 \cdot (- 1)

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 1 \cdot 1

(- 2)^{2} = 2

(- 2)^{2}

使用指数法则:

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

若

n

是偶数

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

= (2)^{2}

使用根式运算法则:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

使用指数法则:

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

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 2

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 4

= 2 + 4

数字相加:

2 + 4 = 6

= 6

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

将解分隔开

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

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

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

使用法则

- (- a) = a

= \frac{2 + 6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{2 + 6}{2}

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

\frac{- ( - 2 ) - 6}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{2 - 6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{2 - 6}{2}

二次方程组的解是:

u = \frac{2 + 6}{2}, u = \frac{2 - 6}{2}

u = \frac{2 + 6}{2}, u = \frac{2 - 6}{2}

验证解

找到无定义的点（奇点）:

u = 0

取

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 = \frac{2 + 6}{2}, u = \frac{2 - 6}{2}

u = \frac{2 + 6}{2}, u = \frac{2 - 6}{2}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{2 + 6}{2} : x = ln (\frac{2 + 6}{2})

e^{x} = \frac{2 + 6}{2}

使用指数运算法则

e^{x} = \frac{2 + 6}{2}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{2 + 6}{2})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{2 + 6}{2})

x = ln (\frac{2 + 6}{2})

解

e^{x} = \frac{2 - 6}{2} : x \in R

无解

e^{x} = \frac{2 - 6}{2}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = ln (\frac{2 + 6}{2})

x = ln (\frac{2 + 6}{2})

流行的例子

sin^2(2x)=cos^2(x)

sin^{2} (2 x) = cos^{2} (x)

-2sin(2x)sin(x)=sin(2x)

- 2 sin (2 x) sin (x) = sin (2 x)

2sin(θ)-sqrt(2)=0,0<= θ<= 2pi

2 sin (θ) - 2 = 0, 0 \leq θ \leq 2 π

sin(θ)-0.2cos(θ)=(6.25)/(9.8)

sin (θ) - 0.2 cos (θ) = \frac{6.25}{9.8}

3cos(θ)=8tan(θ)

3 cos (θ) = 8 tan (θ)