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受欢迎的三角函数 >

1/2 sinh(2x)-4/5 cosh(2x)+1=0

解答

\frac{1}{2} sinh (2 x) - \frac{4}{5} cosh (2 x) + 1 = 0

解答

x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

+1

度数

x = - 9.01906 \dots^{\circ}, x = 51.02652 \dots^{\circ}

求解步骤

\frac{1}{2} sinh (2 x) - \frac{4}{5} cosh (2 x) + 1 = 0

使用三角恒等式改写

\frac{1}{2} sinh (2 x) - \frac{4}{5} cosh (2 x) + 1 = 0

使用双曲函数恒等式:

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

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} cosh (2 x) + 1 = 0

使用双曲函数恒等式:

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

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0 : x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} + 1 = 0

找到

4, 10

的最小公倍数:

20

4, 10

最小公倍数 (LCM)

4

质因数分解:

2 \cdot 2

4

4

除以

24 = 2 \cdot 2

= 2 \cdot 2

10

质因数分解:

2 \cdot 5

10

10

除以

210 = 5 \cdot 2

= 2 \cdot 5

2, 5

都是质数，因此无法进一步因数分解

= 2 \cdot 5

将每个因子乘以它在

4

或

10

中出现的最多次数

= 2 \cdot 2 \cdot 5

数字相乘:

2 \cdot 2 \cdot 5 = 20

= 20

乘以最小公倍数=

20

\frac{1}{2} \cdot \frac{e ^{2 x} - e ^{- 2 x}}{2} \cdot 20 - \frac{4}{5} \cdot \frac{e ^{2 x} + e ^{- 2 x}}{2} \cdot 20 + 1 \cdot 20 = 0 \cdot 20

化简

5 (e^{2 x} - e^{- 2 x}) - 8 (e^{2 x} + e^{- 2 x}) + 20 = 0

使用指数运算法则

5 (e^{2 x} - e^{- 2 x}) - 8 (e^{2 x} + e^{- 2 x}) + 20 = 0

使用指数法则:

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

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

5 ((e^{x})^{2} - (e^{x})^{- 2}) - 8 ((e^{x})^{2} + (e^{x})^{- 2}) + 20 = 0

5 ((e^{x})^{2} - (e^{x})^{- 2}) - 8 ((e^{x})^{2} + (e^{x})^{- 2}) + 20 = 0

用

e^{x} = u

改写方程式

5 ((u)^{2} - (u)^{- 2}) - 8 ((u)^{2} + (u)^{- 2}) + 20 = 0

解

5 (u^{2} - u^{- 2}) - 8 (u^{2} + u^{- 2}) + 20 = 0 : u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

5 (u^{2} - u^{- 2}) - 8 (u^{2} + u^{- 2}) + 20 = 0

整理后得

5 (u^{2} - \frac{1}{u ^{2}}) - 8 (u^{2} + \frac{1}{u ^{2}}) + 20 = 0

展开

5 (u^{2} - \frac{1}{u ^{2}}) - 8 (u^{2} + \frac{1}{u ^{2}}) + 20 : - 3 u^{2} - \frac{13}{u ^{2}} + 20

5 (u^{2} - \frac{1}{u ^{2}}) - 8 (u^{2} + \frac{1}{u ^{2}}) + 20

乘开

5 (u^{2} - \frac{1}{u ^{2}}) : 5 u^{2} - \frac{5}{u ^{2}}

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

使用分配律:

a (b - c) = ab - a c

a = 5, b = u^{2}, c = \frac{1}{u ^{2}}

= 5 u^{2} - 5 \cdot \frac{1}{u ^{2}}

5 \cdot \frac{1}{u ^{2}} = \frac{5}{u ^{2}}

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

分式相乘:

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

= \frac{1 \cdot 5}{u ^{2}}

数字相乘:

1 \cdot 5 = 5

= \frac{5}{u ^{2}}

= 5 u^{2} - \frac{5}{u ^{2}}

= 5 u^{2} - \frac{5}{u ^{2}} - 8 (u^{2} + \frac{1}{u ^{2}}) + 20

乘开

- 8 (u^{2} + \frac{1}{u ^{2}}) : - 8 u^{2} - \frac{8}{u ^{2}}

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

使用分配律:

a (b + c) = ab + a c

a = - 8, b = u^{2}, c = \frac{1}{u ^{2}}

= - 8 u^{2} + (- 8) \frac{1}{u ^{2}}

使用加减运算法则

+ (- a) = - a

= - 8 u^{2} - 8 \cdot \frac{1}{u ^{2}}

8 \cdot \frac{1}{u ^{2}} = \frac{8}{u ^{2}}

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

分式相乘:

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

= \frac{1 \cdot 8}{u ^{2}}

数字相乘:

1 \cdot 8 = 8

= \frac{8}{u ^{2}}

= - 8 u^{2} - \frac{8}{u ^{2}}

= 5 u^{2} - \frac{5}{u ^{2}} - 8 u^{2} - \frac{8}{u ^{2}} + 20

化简

5 u^{2} - \frac{5}{u ^{2}} - 8 u^{2} - \frac{8}{u ^{2}} + 20 : - 3 u^{2} - \frac{13}{u ^{2}} + 20

5 u^{2} - \frac{5}{u ^{2}} - 8 u^{2} - \frac{8}{u ^{2}} + 20

对同类项分组

= 5 u^{2} - 8 u^{2} - \frac{5}{u ^{2}} - \frac{8}{u ^{2}} + 20

合并分式

- \frac{5}{u ^{2}} - \frac{8}{u ^{2}} : - \frac{13}{u ^{2}}

使用法则

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

= \frac{- 5 - 8}{u ^{2}}

数字相减:

- 5 - 8 = - 13

= \frac{- 13}{u ^{2}}

使用分式法则:

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

= - \frac{13}{u ^{2}}

= 5 u^{2} - 8 u^{2} - \frac{13}{u ^{2}} + 20

同类项相加:

5 u^{2} - 8 u^{2} = - 3 u^{2}

= - 3 u^{2} - \frac{13}{u ^{2}} + 20

= - 3 u^{2} - \frac{13}{u ^{2}} + 20

- 3 u^{2} - \frac{13}{u ^{2}} + 20 = 0

在两边乘以

u^{2}

- 3 u^{2} - \frac{13}{u ^{2}} + 20 = 0

在两边乘以

u^{2}

- 3 u^{2} u^{2} - \frac{13}{u ^{2}} u^{2} + 20 u^{2} = 0 \cdot u^{2}

化简

- 3 u^{2} u^{2} - \frac{13}{u ^{2}} u^{2} + 20 u^{2} = 0 \cdot u^{2}

化简

- 3 u^{2} u^{2} : - 3 u^{4}

- 3 u^{2} u^{2}

使用指数法则:

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

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

= - 3 u^{2 + 2}

数字相加:

2 + 2 = 4

= - 3 u^{4}

化简

- \frac{13}{u ^{2}} u^{2} : - 13

- \frac{13}{u ^{2}} u^{2}

分式相乘:

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

= - \frac{13 u ^{2}}{u ^{2}}

约分:

u^{2}

= - 13

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

- 3 u^{4} - 13 + 20 u^{2} = 0

- 3 u^{4} - 13 + 20 u^{2} = 0

- 3 u^{4} - 13 + 20 u^{2} = 0

解

- 3 u^{4} - 13 + 20 u^{2} = 0 : u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

- 3 u^{4} - 13 + 20 u^{2} = 0

改写成标准形式

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

- 3 u^{4} + 20 u^{2} - 13 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

- 3 v^{2} + 20 v - 13 = 0

解

- 3 v^{2} + 20 v - 13 = 0 : v = \frac{10 - 61}{3}, v = \frac{10 + 61}{3}

- 3 v^{2} + 20 v - 13 = 0

使用求根公式求解

- 3 v^{2} + 20 v - 13 = 0

二次方程求根公式:

若

a = - 3, b = 20, c = - 13

v_{1, 2} = \frac{- 20 \pm 2 0 ^{2} - 4 ( - 3 ) ( - 13 )}{2 ( - 3 )}

v_{1, 2} = \frac{- 20 \pm 2 0 ^{2} - 4 ( - 3 ) ( - 13 )}{2 ( - 3 )}

2 0^{2} - 4 (- 3) (- 13) = 261

2 0^{2} - 4 (- 3) (- 13)

使用法则

- (- a) = a

= 2 0^{2} - 4 \cdot 3 \cdot 13

数字相乘:

4 \cdot 3 \cdot 13 = 156

= 2 0^{2} - 156

2 0^{2} = 400

= 400 - 156

数字相减:

400 - 156 = 244

= 244

244

质因数分解:

2^{2} \cdot 61

244

244

除以

2244 = 122 \cdot 2

= 2 \cdot 122

122

除以

2122 = 61 \cdot 2

= 2 \cdot 2 \cdot 61

2, 61

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 61

= 2^{2} \cdot 61

= 2^{2} \cdot 61

使用根式运算法则:

n ab = n a n b

= 61 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 261

v_{1, 2} = \frac{- 20 \pm 2 61}{2 ( - 3 )}

将解分隔开

v_{1} = \frac{- 20 + 2 61}{2 ( - 3 )}, v_{2} = \frac{- 20 - 2 61}{2 ( - 3 )}

v = \frac{- 20 + 2 61}{2 ( - 3 )} : \frac{10 - 61}{3}

\frac{- 20 + 2 61}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 20 + 2 61}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 20 + 2 61}{- 6}

使用分式法则:

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

- 20 + 261 = - (20 - 261)

= \frac{20 - 2 61}{6}

分解

20 - 261 : 2 (10 - 61)

20 - 261

改写为

= 2 \cdot 10 - 261

因式分解出通项

2

= 2 (10 - 61)

= \frac{2 ( 10 - 61 )}{6}

约分:

2

= \frac{10 - 61}{3}

v = \frac{- 20 - 2 61}{2 ( - 3 )} : \frac{10 + 61}{3}

\frac{- 20 - 2 61}{2 ( - 3 )}

去除括号:

(- a) = - a

= \frac{- 20 - 2 61}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{- 20 - 2 61}{- 6}

使用分式法则:

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

- 20 - 261 = - (20 + 261)

= \frac{20 + 2 61}{6}

分解

20 + 261 : 2 (10 + 61)

20 + 261

改写为

= 2 \cdot 10 + 261

因式分解出通项

2

= 2 (10 + 61)

= \frac{2 ( 10 + 61 )}{6}

约分:

2

= \frac{10 + 61}{3}

二次方程组的解是:

v = \frac{10 - 61}{3}, v = \frac{10 + 61}{3}

v = \frac{10 - 61}{3}, v = \frac{10 + 61}{3}

代回

v = u^{2}

，求解

u

解

u^{2} = \frac{10 - 61}{3} : u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}

u^{2} = \frac{10 - 61}{3}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}

解

u^{2} = \frac{10 + 61}{3} : u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

u^{2} = \frac{10 + 61}{3}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

解为

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

验证解

找到无定义的点（奇点）:

u = 0

取

5 (u^{2} - u^{- 2}) - 8 (u^{2} + u^{- 2}) + 20

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

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{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

u = \frac{10 - 61}{3}, u = - \frac{10 - 61}{3}, u = \frac{10 + 61}{3}, u = - \frac{10 + 61}{3}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{10 - 61}{3} : x = \frac{1}{2} ln (\frac{10 - 61}{3})

e^{x} = \frac{10 - 61}{3}

使用指数运算法则

e^{x} = \frac{10 - 61}{3}

使用指数法则:

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

\frac{10 - 61}{3} = (\frac{10 - 61}{3})^{\frac{1}{2}}

e^{x} = (\frac{10 - 61}{3})^{\frac{1}{2}}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{10 - 61}{3})^{\frac{1}{2}}

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{10 - 61}{3})^{\frac{1}{2}}

使用对数计算法则:

ln (x^{a}) = a \cdot ln (x)

ln (\frac{10 - 61}{3})^{\frac{1}{2}} = \frac{1}{2} ln (\frac{10 - 61}{3})

x = \frac{1}{2} ln (\frac{10 - 61}{3})

x = \frac{1}{2} ln (\frac{10 - 61}{3})

解

e^{x} = - \frac{10 - 61}{3} : x \in R

无解

e^{x} = - \frac{10 - 61}{3}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

解

e^{x} = \frac{10 + 61}{3} : x = \frac{1}{2} ln (\frac{10 + 61}{3})

e^{x} = \frac{10 + 61}{3}

使用指数运算法则

e^{x} = \frac{10 + 61}{3}

使用指数法则:

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

\frac{10 + 61}{3} = (\frac{10 + 61}{3})^{\frac{1}{2}}

e^{x} = (\frac{10 + 61}{3})^{\frac{1}{2}}

若

f (x) = g (x)

，则

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

ln (e^{x}) = ln (\frac{10 + 61}{3})^{\frac{1}{2}}

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{10 + 61}{3})^{\frac{1}{2}}

使用对数计算法则:

ln (x^{a}) = a \cdot ln (x)

ln (\frac{10 + 61}{3})^{\frac{1}{2}} = \frac{1}{2} ln (\frac{10 + 61}{3})

x = \frac{1}{2} ln (\frac{10 + 61}{3})

x = \frac{1}{2} ln (\frac{10 + 61}{3})

解

e^{x} = - \frac{10 + 61}{3} : x \in R

无解

e^{x} = - \frac{10 + 61}{3}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

x = \frac{1}{2} ln (\frac{10 - 61}{3}), x = \frac{1}{2} ln (\frac{10 + 61}{3})

作图

流行的例子

7sin^2(x)+2sin(x)-2=0

7 sin^{2} (x) + 2 sin (x) - 2 = 0

3 cos (θ) - 1 = - 1

2 - 4 cos (x) = 0

csc (2 x) - 4 = 0

8cos(2x)=4sqrt(2)

8 cos (2 x) = 42