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sinh(x)=2cosh(x)sinh(x)

解答

sinh (x) = 2 cosh (x) sinh (x)

解答

x = 0

+1

度数

x = 0^{\circ}

求解步骤

sinh (x) = 2 cosh (x) sinh (x)

使用三角恒等式改写

sinh (x) = 2 cosh (x) sinh (x)

使用双曲函数恒等式:

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

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

使用双曲函数恒等式:

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

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

使用双曲函数恒等式:

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

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

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

\frac{e ^{x} - e ^{- x}}{2} = 2 \cdot \frac{e ^{x} + e ^{- x}}{2} \cdot \frac{e ^{x} - e ^{- x}}{2} : x = 0

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

在两边乘以

2

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

化简

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

使用指数运算法则

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

使用指数法则:

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

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

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

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

用

e^{x} = u

改写方程式

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

解

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

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

整理后得

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

在两边乘以

u

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

在两边乘以

u

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

化简

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

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

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

展开

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

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

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

乘开

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

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

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

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

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

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

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

使用指数法则:

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

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

使用法则

1^{a} = 1

1^{2} = 1

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

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

a = u, b = u^{2}, c = \frac{1}{u ^{2}}

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

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

化简

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

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

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

u^{2} u

使用指数法则:

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

u^{2} u = u^{2 + 1}

= u^{2 + 1}

数字相加:

2 + 1 = 3

= u^{3}

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

\frac{1}{u ^{2}} u

分式相乘:

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

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

乘以:

1 \cdot u = u

= \frac{u}{u ^{2}}

约分:

u

= \frac{1}{u}

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

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

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

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

在两边乘以

u

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

在两边乘以

u

u^{2} u - 1 \cdot u = u^{3} u - \frac{1}{u} u

化简

u^{2} u - 1 \cdot u = u^{3} u - \frac{1}{u} u

化简

u^{2} u : u^{3}

u^{2} u

使用指数法则:

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

u^{2} u = u^{2 + 1}

= u^{2 + 1}

数字相加:

2 + 1 = 3

= u^{3}

化简

- 1 \cdot u : - u

- 1 \cdot u

乘以:

1 \cdot u = u

= - u

化简

u^{3} u : u^{4}

u^{3} u

使用指数法则:

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

u^{3} u = u^{3 + 1}

= u^{3 + 1}

数字相加:

3 + 1 = 4

= u^{4}

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^{3} - u = u^{4} - 1

u^{3} - u = u^{4} - 1

u^{3} - u = u^{4} - 1

解

u^{3} - u = u^{4} - 1 : u = 1, u = - 1

u^{3} - u = u^{4} - 1

交换两边

u^{4} - 1 = u^{3} - u

将

u

para o lado esquerdo

u^{4} - 1 = u^{3} - u

两边加上

u

u^{4} - 1 + u = u^{3} - u + u

化简

u^{4} - 1 + u = u^{3}

u^{4} - 1 + u = u^{3}

将

u^{3}

para o lado esquerdo

u^{4} - 1 + u = u^{3}

两边减去

u^{3}

u^{4} - 1 + u - u^{3} = u^{3} - u^{3}

化简

u^{4} - 1 + u - u^{3} = 0

u^{4} - 1 + u - u^{3} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - u^{3} + u - 1 = 0

因式分解

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

u^{4} - u^{3} + u - 1

= (u^{4} - u^{3}) + (u - 1)

从

u^{4} - u^{3}

分解出因式

u^{3}

:

u^{3} (u - 1)

u^{4} - u^{3}

使用指数法则:

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

u^{4} = u u^{3}

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

因式分解出通项

u^{3}

= u^{3} (u - 1)

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

因式分解出通项

u - 1

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

分解

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

u^{3} + 1

将

1

改写为

1^{3}

= u^{3} + 1^{3}

使用立方和公式:

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

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

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

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

(u - 1) (u + 1) (u^{2} - u + 1) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

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

解

u - 1 = 0 : u = 1

u - 1 = 0

将

1

到右边

u - 1 = 0

两边加上

1

u - 1 + 1 = 0 + 1

化简

u = 1

u = 1

解

u + 1 = 0 : u = - 1

u + 1 = 0

将

1

到右边

u + 1 = 0

两边减去

1

u + 1 - 1 = 0 - 1

化简

u = - 1

u = - 1

解

u^{2} - u + 1 = 0 : u \in R

无解

u^{2} - u + 1 = 0

判别式

u^{2} - u + 1 = 0 : - 3

u^{2} - u + 1 = 0

对于

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

形式的二次方程，根判别式为

b^{2} - 4 a c

若

a = 1, b = - 1, c = 1 : (- 1)^{2} - 4 \cdot 1 \cdot 1

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

展开

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

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

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 4

= 1 - 4

数字相减:

1 - 4 = - 3

= - 3

- 3

判别式在

u

内不能为负

\in R

解是

u \in R 无解

解为

u = 1, u = - 1

u = 1, u = - 1

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

取

(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 = 1, u = - 1

u = 1, u = - 1

代回

u = e^{x}

，求解

x

解

e^{x} = 1 : x = 0

e^{x} = 1

使用指数运算法则

e^{x} = 1

若

f (x) = g (x)

，则

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

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

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1)

化简

ln (1) : 0

ln (1)

使用对数计算法则:

lo g_{a} (1) = 0

= 0

x = 0

x = 0

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 = 0

x = 0

流行的例子

cos^2(x)=sin(x)+1

cos^{2} (x) = sin (x) + 1

tan (x) = \frac{5}{- 4}

cot (α) = 4.9

0 = 8 tan (θ)

1 = 4 sin^{2} (θ)