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受欢迎的三角函数 >

cosh(x-1)=2

解答

cosh (x - 1) = 2

解答

x = ln (e (2 + 3)), x = ln (e (2 - 3))

+1

度数

x = 132.75190 \dots^{\circ}, x = - 18.16034 \dots^{\circ}

求解步骤

cosh (x - 1) = 2

使用三角恒等式改写

cosh (x - 1) = 2

使用双曲函数恒等式:

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

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

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

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

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

在两边乘以

2

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

化简

e^{x - 1} + e^{- (x - 1)} = 4

使用指数运算法则

e^{x - 1} + e^{- (x - 1)} = 4

使用指数法则:

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

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

e^{x} e^{- 1} + e^{- 1 \cdot x} e^{1} = 4

使用指数法则:

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

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

e^{x} e^{- 1} + (e^{x})^{- 1} e^{1} = 4

e^{x} e^{- 1} + (e^{x})^{- 1} e^{1} = 4

用

e^{x} = u

改写方程式

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

解

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

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

整理后得

\frac{1}{e} u + \frac{e}{u} = 4

乘以最小公倍数

\frac{1}{e} u + \frac{e}{u} = 4

找到

e, u

的最小公倍数:

e u

e, u

最小公倍数 (LCM)

计算出由出现在

e

或

u

中的因子组成的表达式

= e u

乘以最小公倍数=

e u

\frac{1}{e} u e u + \frac{e}{u} e u = 4 e u

化简

\frac{1}{e} u e u + \frac{e}{u} e u = 4 e u

化简

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

\frac{1}{e} u e u

使用指数法则:

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

uu = u^{1 + 1}

= \frac{1}{e} e u^{1 + 1}

数字相加:

1 + 1 = 2

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

分式相乘:

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

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

约分:

e

= u^{2} \cdot 1

乘以:

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

= u^{2}

化简

\frac{e}{u} e u : e^{2}

\frac{e}{u} e u

分式相乘:

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

= \frac{ee u}{u}

约分:

u

= ee

使用指数法则:

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

ee = e^{1 + 1}

= e^{1 + 1}

数字相加:

1 + 1 = 2

= e^{2}

u^{2} + e^{2} = 4 e u

u^{2} + e^{2} = 4 e u

u^{2} + e^{2} = 4 e u

解

u^{2} + e^{2} = 4 e u : u = e (2 + 3), u = e (2 - 3)

u^{2} + e^{2} = 4 e u

将

4 e u

para o lado esquerdo

u^{2} + e^{2} = 4 e u

两边减去

4 e u

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

化简

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = - 4 e, c = e^{2}

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

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

(- 4 e)^{2} - 4 \cdot 1 \cdot e^{2} = 23 e

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

(- 4 e)^{2} = 4^{2} e^{2}

(- 4 e)^{2}

使用指数法则:

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

若

n

是偶数

(- 4 e)^{2} = (4 e)^{2}

= (4 e)^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} e^{2}

4 \cdot 1 \cdot e^{2} = 4 e^{2}

4 \cdot 1 \cdot e^{2}

数字相乘:

4 \cdot 1 = 4

= 4 e^{2}

= 4^{2} e^{2} - 4 e^{2}

4^{2} = 16

= 16 e^{2} - 4 e^{2}

同类项相加:

16 e^{2} - 4 e^{2} = 12 e^{2}

= 12 e^{2}

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= 12 e^{2}

12 = 23

12

12

质因数分解:

2^{2} \cdot 3

12

12

除以

212 = 6 \cdot 2

= 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

使用根式运算法则:

n ab = n a n b

= 3 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 23

= 23 e^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

e^{2} = e

= 23 e

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

将解分隔开

u_{1} = \frac{- ( - 4 e ) + 2 3 e}{2 \cdot 1}, u_{2} = \frac{- ( - 4 e ) - 2 3 e}{2 \cdot 1}

u = \frac{- ( - 4 e ) + 2 3 e}{2 \cdot 1} : e (2 + 3)

\frac{- ( - 4 e ) + 2 3 e}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{4 e + 2 3 e}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{4 e + 2 3 e}{2}

分解

4 e + 23 e : 2 e (2 + 3)

4 e + 23 e

改写为

= 2 \cdot 2 e + 2 e 3

因式分解出通项

2 e

= 2 e (2 + 3)

= \frac{2 e ( 2 + 3 )}{2}

数字相除:

\frac{2}{2} = 1

= e (2 + 3)

u = \frac{- ( - 4 e ) - 2 3 e}{2 \cdot 1} : e (2 - 3)

\frac{- ( - 4 e ) - 2 3 e}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{4 e - 2 3 e}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

分解

4 e - 23 e : 2 e (2 - 3)

4 e - 23 e

改写为

= 2 \cdot 2 e - 2 e 3

因式分解出通项

2 e

= 2 e (2 - 3)

= \frac{2 e ( 2 - 3 )}{2}

数字相除:

\frac{2}{2} = 1

= e (2 - 3)

二次方程组的解是:

u = e (2 + 3), u = e (2 - 3)

u = e (2 + 3), u = e (2 - 3)

验证解

找到无定义的点（奇点）:

u = 0

取

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

u = e (2 + 3), u = e (2 - 3)

代回

u = e^{x}

，求解

x

解

e^{x} = e (2 + 3) : x = ln (e (2 + 3))

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

使用指数运算法则

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

若

f (x) = g (x)

，则

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

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

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (e (2 + 3))

x = ln (e (2 + 3))

解

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

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

使用指数运算法则

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

若

f (x) = g (x)

，则

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

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

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (e (2 - 3))

x = ln (e (2 - 3))

x = ln (e (2 + 3)), x = ln (e (2 - 3))

x = ln (e (2 + 3)), x = ln (e (2 - 3))

作图

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