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3-cosh(x)=0

解答

3 - cosh (x) = 0

解答

x = ln (3 - 22), x = ln (3 + 22)

+1

度数

x = - 100.99797 \dots^{\circ}, x = 100.99797 \dots^{\circ}

求解步骤

3 - cosh (x) = 0

使用三角恒等式改写

3 - cosh (x) = 0

使用双曲函数恒等式:

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

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

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

3 - \frac{e ^{x} + e ^{- x}}{2} = 0 : x = ln (3 - 22), x = ln (3 + 22)

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

在两边乘以

2

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

化简

6 - (e^{x} + e^{- x}) = 0

两边加上

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

6 - (e^{x} + e^{- x}) + e^{x} + e^{- x} = 0 + e^{x} + e^{- x}

化简

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

使用指数运算法则

6 - (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}

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

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

用

e^{x} = u

改写方程式

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

解

6 - (u + u^{- 1}) + u + u^{- 1} = u + u^{- 1} : u = 3 - 22, u = 3 + 22

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

整理后得

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

两边减去

u + \frac{1}{u}

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

化简

6 - (u + \frac{1}{u}) = 0

化简

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

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

打开括号

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

使用加减运算法则

+ (- a) = - a

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

6 - u - \frac{1}{u} = 0

在两边乘以

u

6 - u - \frac{1}{u} = 0

在两边乘以

u

6 u - uu - \frac{1}{u} u = 0 \cdot u

化简

6 u - uu - \frac{1}{u} u = 0 \cdot 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

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

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

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

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

解

6 u - u^{2} - 1 = 0 : u = 3 - 22, u = 3 + 22

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

改写成标准形式

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

- u^{2} + 6 u - 1 = 0

使用求根公式求解

- u^{2} + 6 u - 1 = 0

二次方程求根公式:

若

a = - 1, b = 6, c = - 1

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) ( - 1 )}{2 ( - 1 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) ( - 1 )}{2 ( - 1 )}

6^{2} - 4 (- 1) (- 1) = 42

6^{2} - 4 (- 1) (- 1)

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 6^{2} - 4

6^{2} = 36

= 36 - 4

数字相减:

36 - 4 = 32

= 32

32

质因数分解:

2^{5}

32

32

除以

232 = 16 \cdot 2

= 2 \cdot 16

16

除以

216 = 8 \cdot 2

= 2 \cdot 2 \cdot 8

8

除以

28 = 4 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

= 2^{5}

= 2^{5}

使用指数法则:

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

= 2^{4} \cdot 2

使用根式运算法则:

= 2 2^{4}

使用根式运算法则:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2

整理后得

= 42

u_{1, 2} = \frac{- 6 \pm 4 2}{2 ( - 1 )}

将解分隔开

u_{1} = \frac{- 6 + 4 2}{2 ( - 1 )}, u_{2} = \frac{- 6 - 4 2}{2 ( - 1 )}

u = \frac{- 6 + 4 2}{2 ( - 1 )} : 3 - 22

\frac{- 6 + 4 2}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 6 + 4 2}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 + 4 2}{- 2}

使用分式法则:

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

- 6 + 42 = - (6 - 42)

= \frac{6 - 4 2}{2}

分解

6 - 42 : 2 (3 - 22)

6 - 42

改写为

= 2 \cdot 3 - 2 \cdot 22

因式分解出通项

2

= 2 (3 - 22)

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

数字相除:

\frac{2}{2} = 1

= 3 - 22

u = \frac{- 6 - 4 2}{2 ( - 1 )} : 3 + 22

\frac{- 6 - 4 2}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 6 - 4 2}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 - 4 2}{- 2}

使用分式法则:

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

- 6 - 42 = - (6 + 42)

= \frac{6 + 4 2}{2}

分解

6 + 42 : 2 (3 + 22)

6 + 42

改写为

= 2 \cdot 3 + 2 \cdot 22

因式分解出通项

2

= 2 (3 + 22)

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

数字相除:

\frac{2}{2} = 1

= 3 + 22

二次方程组的解是:

u = 3 - 22, u = 3 + 22

u = 3 - 22, u = 3 + 22

验证解

找到无定义的点（奇点）:

u = 0

取

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

的分母，令其等于零

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 = 3 - 22, u = 3 + 22

u = 3 - 22, u = 3 + 22

代回

u = e^{x}

，求解

x

解

e^{x} = 3 - 22 : x = ln (3 - 22)

e^{x} = 3 - 22

使用指数运算法则

e^{x} = 3 - 22

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (3 - 22)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3 - 22)

x = ln (3 - 22)

解

e^{x} = 3 + 22 : x = ln (3 + 22)

e^{x} = 3 + 22

使用指数运算法则

e^{x} = 3 + 22

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (3 + 22)

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3 + 22)

x = ln (3 + 22)

x = ln (3 - 22), x = ln (3 + 22)

x = ln (3 - 22), x = ln (3 + 22)

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sin(x)tan(x)+sec(x)+5=0

sin (x) tan (x) + sec (x) + 5 = 0

8 - 32 cos^{2} (t) = 0

tan (x) = \frac{12}{9}

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