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受欢迎的三角函数 >

arctan(x/(12))-arctan(x)=0.001

解答

arctan (\frac{x}{12}) - arctan (x) = 0.001

解答

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

求解步骤

arctan (\frac{x}{12}) - arctan (x) = 0.001

使用三角恒等式改写

arctan (\frac{x}{12}) - arctan (x)

使用和差化积恒等式:

arctan (s) - arctan (t) = arctan (\frac{s - t}{1 + s t})

= arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x})

arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}) = 0.001

使用反三角函数性质

arctan (\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}) = 0.001

arctan (x) = a \Rightarrow x = tan (a)

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (0.001)

tan (0.001) = tan (\frac{1}{1000})

tan (0.001)

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

解

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000}) : x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} = tan (\frac{1}{1000})

化简

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x} : - \frac{11 x}{12 + x ^{2}}

\frac{\frac{x}{12} - x}{1 + \frac{x}{12} x}

\frac{x}{12} x = \frac{x ^{2}}{12}

\frac{x}{12} x

分式相乘:

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

= \frac{xx}{12}

xx = x^{2}

xx

使用指数法则:

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

xx = x^{1 + 1}

= x^{1 + 1}

数字相加:

1 + 1 = 2

= x^{2}

= \frac{x ^{2}}{12}

= \frac{\frac{x}{12} - x}{1 + \frac{x ^{2}}{12}}

化简

\frac{x}{12} - x : - \frac{11 x}{12}

\frac{x}{12} - x

将项转换为分式:

x = \frac{x 12}{12}

= \frac{x}{12} - \frac{x \cdot 12}{12}

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{x - x \cdot 12}{12}

同类项相加:

x - 12 x = - 11 x

= \frac{- 11 x}{12}

使用分式法则:

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

= - \frac{11 x}{12}

= \frac{- \frac{11 x}{12}}{1 + \frac{x ^{2}}{12}}

使用分式法则:

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

= - \frac{\frac{11 x}{12}}{1 + \frac{x ^{2}}{12}}

使用分式法则:

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

\frac{\frac{11 x}{12}}{1 + \frac{x ^{2}}{12}} = \frac{11 x}{12 ( 1 + \frac{x ^{2}}{12} )}

= - \frac{11 x}{12 ( 1 + \frac{x ^{2}}{12} )}

化简

1 + \frac{x ^{2}}{12} : \frac{12 + x ^{2}}{12}

1 + \frac{x ^{2}}{12}

将项转换为分式:

1 = \frac{1 \cdot 12}{12}

= \frac{1 \cdot 12}{12} + \frac{x ^{2}}{12}

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 12 + x ^{2}}{12}

数字相乘:

1 \cdot 12 = 12

= \frac{12 + x ^{2}}{12}

= - \frac{11 x}{12 \cdot \frac{x ^{2} + 12}{12}}

乘

12 \cdot \frac{12 + x ^{2}}{12} : 12 + x^{2}

12 \cdot \frac{12 + x ^{2}}{12}

分式相乘:

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

= \frac{( 12 + x ^{2} ) \cdot 12}{12}

约分:

12

= 12 + x^{2}

= - \frac{11 x}{x ^{2} + 12}

- \frac{11 x}{12 + x ^{2}} = tan (\frac{1}{1000})

在两边乘以

12 + x^{2}

- \frac{11 x}{12 + x ^{2}} = tan (\frac{1}{1000})

在两边乘以

12 + x^{2}

- \frac{11 x}{12 + x ^{2}} (12 + x^{2}) = tan (\frac{1}{1000}) (12 + x^{2})

化简

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

解

- 11 x = tan (\frac{1}{1000}) (12 + x^{2}) : x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

- 11 x = tan (\frac{1}{1000}) (12 + x^{2})

展开

tan (\frac{1}{1000}) (12 + x^{2}) : 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

tan (\frac{1}{1000}) (12 + x^{2})

使用分配律:

a (b + c) = ab + a c

a = tan (\frac{1}{1000}), b = 12, c = x^{2}

= tan (\frac{1}{1000}) \cdot 12 + tan (\frac{1}{1000}) x^{2}

= 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

- 11 x = 12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2}

交换两边

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} = - 11 x

将

11 x

para o lado esquerdo

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} = - 11 x

两边加上

11 x

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = - 11 x + 11 x

化简

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = 0

12 tan (\frac{1}{1000}) + tan (\frac{1}{1000}) x^{2} + 11 x = 0

改写成标准形式

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

tan (\frac{1}{1000}) x^{2} + 11 x + 12 tan (\frac{1}{1000}) = 0

使用求根公式求解

tan (\frac{1}{1000}) x^{2} + 11 x + 12 tan (\frac{1}{1000}) = 0

二次方程求根公式:

若

a = tan (\frac{1}{1000}), b = 11, c = 12 tan (\frac{1}{1000})

x_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 tan ( \frac{1}{1000} ) \cdot 12 tan ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 tan ( \frac{1}{1000} ) \cdot 12 tan ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

1 1^{2} - 4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000}) = 121 - 48 tan^{2} (\frac{1}{1000})

1 1^{2} - 4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000})

4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000}) = 48 tan^{2} (\frac{1}{1000})

4 tan (\frac{1}{1000}) \cdot 12 tan (\frac{1}{1000})

数字相乘:

4 \cdot 12 = 48

= 48 tan (\frac{1}{1000}) tan (\frac{1}{1000})

使用指数法则:

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

tan (\frac{1}{1000}) tan (\frac{1}{1000}) = tan^{1 + 1} (\frac{1}{1000})

= 48 tan^{1 + 1} (\frac{1}{1000})

数字相加:

1 + 1 = 2

= 48 tan^{2} (\frac{1}{1000})

= 1 1^{2} - 48 tan^{2} (\frac{1}{1000})

1 1^{2} = 121

= 121 - 48 tan^{2} (\frac{1}{1000})

x_{1, 2} = \frac{- 11 \pm 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

将解分隔开

x_{1} = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x_{2} = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

二次方程组的解是:

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

将解代入原方程进行验证

将它们代入

arctan (\frac{x}{12}) - arctan (x) = 0.001

检验解是否符合

去除与方程不符的解。

检验

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

的解:

真

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

代入

n = 1

\frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

对于

arctan (\frac{x}{12}) - arctan (x) = 0.001

代入

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

arctan \frac{\frac{- 11 + 121 - 48 t a n ^{2} ( \frac{1}{1000} )}{2 t a n ( \frac{1}{1000} )}}{12} - arctan \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} = 0.001

整理后得

0.00099 \dots = 0.001

\Rightarrow 真

检验

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

的解:

真

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

代入

n = 1

\frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

对于

arctan (\frac{x}{12}) - arctan (x) = 0.001

代入

x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

arctan \frac{\frac{- 11 - 121 - 48 t a n ^{2} ( \frac{1}{1000} )}{2 t a n ( \frac{1}{1000} )}}{12} - arctan \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )} = 0.001

整理后得

0.001 = 0.001

\Rightarrow 真

x = \frac{- 11 + 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}, x = \frac{- 11 - 121 - 48 tan ^{2} ( \frac{1}{1000} )}{2 tan ( \frac{1}{1000} )}

作图

流行的例子

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