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受欢迎的三角函数 >

arctan(x/3)+arctan(x/2)=arctan(x)

解答

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

解答

x = 0, x = - 1, x = 1

求解步骤

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

两边减去

arctan (x)

arctan (\frac{x}{3}) + arctan (\frac{x}{2}) - arctan (x) = 0

使用三角恒等式改写

- arctan (x) + arctan (\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}})

使用和差化积恒等式:

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

= arctan \frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x}

arctan \frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

使用反三角函数性质

arctan \frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

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

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = tan (0)

tan (0) = 0

tan (0)

使用以下普通恒等式:

tan (0) = 0

tan (0)

tan (x)

周期表（周期为

πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 0

= 0

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

解

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0 : x = 0, x = - 1, x = 1

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} = 0

化简

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x} : \frac{- x + x ^{3}}{6 + 4 x ^{2}}

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x = \frac{5 x ^{2}}{6 - x ^{2}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} = \frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}}

\frac{x}{3} \cdot \frac{x}{2} = \frac{x ^{2}}{6}

\frac{x}{3} \cdot \frac{x}{2}

分式相乘:

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

= \frac{xx}{3 \cdot 2}

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}}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{x ^{2}}{6}

= \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x ^{2}}{6}}

化简

\frac{x}{3} + \frac{x}{2} : \frac{5 x}{6}

\frac{x}{3} + \frac{x}{2}

3, 2

的最小公倍数:

6

3, 2

最小公倍数 (LCM)

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

2

质因数分解:

2

2

2

是质数，因此无法因数分解

= 2

将每个因子乘以它在

3

或

2

中出现的最多次数

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

6

对于

\frac{x}{3} :

将分母和分子乘以

2

\frac{x}{3} = \frac{x \cdot 2}{3 \cdot 2} = \frac{x \cdot 2}{6}

对于

\frac{x}{2} :

将分母和分子乘以

3

\frac{x}{2} = \frac{x \cdot 3}{2 \cdot 3} = \frac{x \cdot 3}{6}

= \frac{x \cdot 2}{6} + \frac{x \cdot 3}{6}

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 \cdot 2 + x \cdot 3}{6}

同类项相加:

2 x + 3 x = 5 x

= \frac{5 x}{6}

= \frac{\frac{5 x}{6}}{1 - \frac{x ^{2}}{6}}

使用分式法则:

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

= \frac{5 x}{6 ( 1 - \frac{x ^{2}}{6} )}

化简

1 - \frac{x ^{2}}{6} : \frac{6 - x ^{2}}{6}

1 - \frac{x ^{2}}{6}

将项转换为分式:

1 = \frac{1 \cdot 6}{6}

= \frac{1 \cdot 6}{6} - \frac{x ^{2}}{6}

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 6 - x ^{2}}{6}

数字相乘:

1 \cdot 6 = 6

= \frac{6 - x ^{2}}{6}

= \frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}}

= \frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}} x

分式相乘:

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

= \frac{5 xx}{6 \cdot \frac{6 - x ^{2}}{6}}

5 xx = 5 x^{2}

5 xx

使用指数法则:

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

xx = x^{1 + 1}

= 5 x^{1 + 1}

数字相加:

1 + 1 = 2

= 5 x^{2}

= \frac{5 x ^{2}}{6 \cdot \frac{- x ^{2} + 6}{6}}

乘

6 \cdot \frac{6 - x ^{2}}{6} : 6 - x^{2}

6 \cdot \frac{6 - x ^{2}}{6}

分式相乘:

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

= \frac{( 6 - x ^{2} ) \cdot 6}{6}

约分:

6

= 6 - x^{2}

= \frac{5 x ^{2}}{6 - x ^{2}}

= \frac{\frac{\frac{x}{3} + \frac{x}{2}}{- \frac{x}{3} \cdot \frac{x}{2} + 1} - x}{1 + \frac{5 x ^{2}}{- x ^{2} + 6}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} = \frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}}

\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}}

\frac{x}{3} \cdot \frac{x}{2} = \frac{x ^{2}}{6}

\frac{x}{3} \cdot \frac{x}{2}

分式相乘:

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

= \frac{xx}{3 \cdot 2}

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}}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{x ^{2}}{6}

= \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x ^{2}}{6}}

化简

\frac{x}{3} + \frac{x}{2} : \frac{5 x}{6}

\frac{x}{3} + \frac{x}{2}

3, 2

的最小公倍数:

6

3, 2

最小公倍数 (LCM)

3

质因数分解:

3

3

3

是质数，因此无法因数分解

= 3

2

质因数分解:

2

2

2

是质数，因此无法因数分解

= 2

将每个因子乘以它在

3

或

2

中出现的最多次数

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

6

对于

\frac{x}{3} :

将分母和分子乘以

2

\frac{x}{3} = \frac{x \cdot 2}{3 \cdot 2} = \frac{x \cdot 2}{6}

对于

\frac{x}{2} :

将分母和分子乘以

3

\frac{x}{2} = \frac{x \cdot 3}{2 \cdot 3} = \frac{x \cdot 3}{6}

= \frac{x \cdot 2}{6} + \frac{x \cdot 3}{6}

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 \cdot 2 + x \cdot 3}{6}

同类项相加:

2 x + 3 x = 5 x

= \frac{5 x}{6}

= \frac{\frac{5 x}{6}}{1 - \frac{x ^{2}}{6}}

使用分式法则:

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

= \frac{5 x}{6 ( 1 - \frac{x ^{2}}{6} )}

化简

1 - \frac{x ^{2}}{6} : \frac{6 - x ^{2}}{6}

1 - \frac{x ^{2}}{6}

将项转换为分式:

1 = \frac{1 \cdot 6}{6}

= \frac{1 \cdot 6}{6} - \frac{x ^{2}}{6}

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 6 - x ^{2}}{6}

数字相乘:

1 \cdot 6 = 6

= \frac{6 - x ^{2}}{6}

= \frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}}

= \frac{\frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}} - x}{1 + \frac{5 x ^{2}}{- x ^{2} + 6}}

化简

1 + \frac{5 x ^{2}}{6 - x ^{2}} : \frac{6 + 4 x ^{2}}{6 - x ^{2}}

1 + \frac{5 x ^{2}}{6 - x ^{2}}

将项转换为分式:

1 = \frac{1 ( 6 - x ^{2} )}{6 - x ^{2}}

= \frac{1 \cdot ( 6 - x ^{2} )}{6 - x ^{2}} + \frac{5 x ^{2}}{6 - x ^{2}}

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 ( 6 - x ^{2} ) + 5 x ^{2}}{6 - x ^{2}}

1 \cdot (6 - x^{2}) + 5 x^{2} = 6 + 4 x^{2}

1 \cdot (6 - x^{2}) + 5 x^{2}

1 \cdot (6 - x^{2}) = 6 - x^{2}

1 \cdot (6 - x^{2})

乘以:

1 \cdot (6 - x^{2}) = (6 - x^{2})

= (6 - x^{2})

去除括号:

(a) = a

= 6 - x^{2}

= 6 - x^{2} + 5 x^{2}

同类项相加:

- x^{2} + 5 x^{2} = 4 x^{2}

= 6 + 4 x^{2}

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

= \frac{\frac{5 x}{6 \cdot \frac{- x ^{2} + 6}{6}} - x}{\frac{6 + 4 x ^{2}}{6 - x ^{2}}}

化简

\frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}} - x : \frac{- x + x ^{3}}{6 - x ^{2}}

\frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}} - x

将项转换为分式:

x = \frac{x 6 \frac{6 - x ^{2}}{6}}{6 \frac{6 - x ^{2}}{6}}

= \frac{5 x}{6 \cdot \frac{6 - x ^{2}}{6}} - \frac{x \cdot 6 \cdot \frac{6 - x ^{2}}{6}}{6 \cdot \frac{6 - x ^{2}}{6}}

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{5 x - x \cdot 6 \cdot \frac{6 - x ^{2}}{6}}{6 \cdot \frac{6 - x ^{2}}{6}}

乘

6 \cdot \frac{6 - x ^{2}}{6} : 6 - x^{2}

6 \cdot \frac{6 - x ^{2}}{6}

分式相乘:

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

= \frac{( 6 - x ^{2} ) \cdot 6}{6}

约分:

6

= 6 - x^{2}

= \frac{5 x - 6 \cdot \frac{- x ^{2} + 6}{6} x}{6 - x ^{2}}

x \cdot 6 \cdot \frac{6 - x ^{2}}{6} = x (6 - x^{2})

x \cdot 6 \cdot \frac{6 - x ^{2}}{6}

分式相乘:

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

= \frac{( 6 - x ^{2} ) x \cdot 6}{6}

约分:

6

= (6 - x^{2}) x

= \frac{5 x - x ( - x ^{2} + 6 )}{6 - x ^{2}}

乘开

5 x - (6 - x^{2}) x : - x + x^{3}

5 x - (6 - x^{2}) x

= 5 x - x (6 - x^{2})

乘开

- x (6 - x^{2}) : - 6 x + x^{3}

- x (6 - x^{2})

使用分配律:

a (b - c) = ab - a c

a = - x, b = 6, c = x^{2}

= - x \cdot 6 - (- x) x^{2}

使用加减运算法则

- (- a) = a

= - 6 x + x^{2} x

x^{2} x = x^{3}

x^{2} x

使用指数法则:

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

x^{2} x = x^{2 + 1}

= x^{2 + 1}

数字相加:

2 + 1 = 3

= x^{3}

= - 6 x + x^{3}

= 5 x - 6 x + x^{3}

同类项相加:

5 x - 6 x = - x

= - x + x^{3}

= \frac{- x + x ^{3}}{6 - x ^{2}}

= \frac{\frac{- x + x ^{3}}{6 - x ^{2}}}{\frac{6 + 4 x ^{2}}{6 - x ^{2}}}

分式相除:

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

= \frac{( - x + x ^{3} ) ( 6 - x ^{2} )}{( 6 - x ^{2} ) ( 6 + 4 x ^{2} )}

约分:

6 - x^{2}

= \frac{- x + x ^{3}}{6 + 4 x ^{2}}

\frac{- x + x ^{3}}{6 + 4 x ^{2}} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- x + x^{3} = 0

解

- x + x^{3} = 0 : x = 0, x = - 1, x = 1

- x + x^{3} = 0

因式分解

- x + x^{3} : x (x + 1) (x - 1)

- x + x^{3}

因式分解出通项

x : x (x^{2} - 1)

x^{3} - x

使用指数法则:

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

x^{3} = x^{2} x

= x^{2} x - x

因式分解出通项

x

= x (x^{2} - 1)

= x (x^{2} - 1)

分解

x^{2} - 1 : (x + 1) (x - 1)

x^{2} - 1

将

1

改写为

1^{2}

= x^{2} - 1^{2}

使用平方差公式:

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

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

= (x + 1) (x - 1)

= x (x + 1) (x - 1)

x (x + 1) (x - 1) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

x = 0 or x + 1 = 0 or x - 1 = 0

解

x + 1 = 0 : x = - 1

x + 1 = 0

将

1

到右边

x + 1 = 0

两边减去

1

x + 1 - 1 = 0 - 1

化简

x = - 1

x = - 1

解

x - 1 = 0 : x = 1

x - 1 = 0

将

1

到右边

x - 1 = 0

两边加上

1

x - 1 + 1 = 0 + 1

化简

x = 1

x = 1

解为

x = 0, x = - 1, x = 1

x = 0, x = - 1, x = 1

验证解

找到无定义的点（奇点）:

x = 6, x = - 6

取

\frac{\frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} - x}{1 + \frac{\frac{x}{3} + \frac{x}{2}}{1 - \frac{x}{3} \cdot \frac{x}{2}} x}

的分母，令其等于零

解

1 - \frac{x}{3} \cdot \frac{x}{2} = 0 : x = 6, x = - 6

1 - \frac{x}{3} \cdot \frac{x}{2} = 0

将

1

到右边

1 - \frac{x}{3} \cdot \frac{x}{2} = 0

两边减去

1

1 - \frac{x}{3} \cdot \frac{x}{2} - 1 = 0 - 1

化简

- \frac{x}{3} \cdot \frac{x}{2} = - 1

- \frac{x}{3} \cdot \frac{x}{2} = - 1

化简

- \frac{x ^{2}}{6} = - 1

在两边乘以

- 6

(- \frac{x ^{2}}{6}) (- 6) = (- 1) (- 6)

x^{2} = 6

对于

x^{2} = f (a)

解为

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

x = 6, x = - 6

以下点无定义

x = 6, x = - 6

将不在定义域的点与解相综合:

x = 0, x = - 1, x = 1

x = 0, x = - 1, x = 1

将解代入原方程进行验证

将它们代入

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

检验解是否符合

去除与方程不符的解。

检验

0

的解:

真

0

代入

n = 1

0

对于

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

代入

x = 0

arctan (\frac{0}{3}) + arctan (\frac{0}{2}) = arctan (0)

整理后得

0 = 0

\Rightarrow 真

检验

- 1

的解:

真

- 1

代入

n = 1

- 1

对于

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

代入

x = - 1

arctan (\frac{- 1}{3}) + arctan (\frac{- 1}{2}) = arctan (- 1)

整理后得

- 0.78539 \dots = - 0.78539 \dots

\Rightarrow 真

检验

1

的解:

真

1

代入

n = 1

1

对于

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

代入

x = 1

arctan (\frac{1}{3}) + arctan (\frac{1}{2}) = arctan (1)

整理后得

0.78539 \dots = 0.78539 \dots

\Rightarrow 真

x = 0, x = - 1, x = 1

作图

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