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受欢迎的三角函数 >

arctan(3x)+arctan(x)= pi/4

解答

arctan (3 x) + arctan (x) = \frac{π}{4}

解答

x = \frac{7 - 2}{3}

求解步骤

arctan (3 x) + arctan (x) = \frac{π}{4}

使用三角恒等式改写

arctan (3 x) + arctan (x)

使用和差化积恒等式:

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

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

arctan (\frac{3 x + x}{1 - 3 xx}) = \frac{π}{4}

使用反三角函数性质

arctan (\frac{3 x + x}{1 - 3 xx}) = \frac{π}{4}

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

\frac{3 x + x}{1 - 3 xx} = tan (\frac{π}{4})

tan (\frac{π}{4}) = 1

tan (\frac{π}{4})

使用以下普通恒等式:

tan (\frac{π}{4}) = 1

tan (\frac{π}{4})

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}

= 1

= 1

\frac{3 x + x}{1 - 3 xx} = 1

\frac{3 x + x}{1 - 3 xx} = 1

解

\frac{3 x + x}{1 - 3 xx} = 1 : x = - \frac{2 + 7}{3}, x = \frac{7 - 2}{3}

\frac{3 x + x}{1 - 3 xx} = 1

化简

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

\frac{3 x + x}{1 - 3 xx}

同类项相加:

3 x + x = 4 x

= \frac{4 x}{1 - 3 xx}

1 - 3 xx = 1 - 3 x^{2}

1 - 3 xx

3 xx = 3 x^{2}

3 xx

使用指数法则:

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

xx = x^{1 + 1}

= 3 x^{1 + 1}

数字相加:

1 + 1 = 2

= 3 x^{2}

= 1 - 3 x^{2}

= \frac{4 x}{1 - 3 x ^{2}}

\frac{4 x}{1 - 3 x ^{2}} = 1

在两边乘以

1 - 3 x^{2}

\frac{4 x}{1 - 3 x ^{2}} = 1

在两边乘以

1 - 3 x^{2}

\frac{4 x}{1 - 3 x ^{2}} (1 - 3 x^{2}) = 1 \cdot (1 - 3 x^{2})

化简

\frac{4 x}{1 - 3 x ^{2}} (1 - 3 x^{2}) = 1 \cdot (1 - 3 x^{2})

化简

\frac{4 x}{1 - 3 x ^{2}} (1 - 3 x^{2}) : 4 x

\frac{4 x}{1 - 3 x ^{2}} (1 - 3 x^{2})

分式相乘:

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

= \frac{4 x ( 1 - 3 x ^{2} )}{1 - 3 x ^{2}}

约分:

1 - 3 x^{2}

= 4 x

化简

1 \cdot (1 - 3 x^{2}) : 1 - 3 x^{2}

1 \cdot (1 - 3 x^{2})

乘以:

1 \cdot (1 - 3 x^{2}) = (1 - 3 x^{2})

= (1 - 3 x^{2})

去除括号:

(a) = a

= 1 - 3 x^{2}

4 x = 1 - 3 x^{2}

4 x = 1 - 3 x^{2}

4 x = 1 - 3 x^{2}

解

4 x = 1 - 3 x^{2} : x = - \frac{2 + 7}{3}, x = \frac{7 - 2}{3}

4 x = 1 - 3 x^{2}

交换两边

1 - 3 x^{2} = 4 x

将

4 x

para o lado esquerdo

1 - 3 x^{2} = 4 x

两边减去

4 x

1 - 3 x^{2} - 4 x = 4 x - 4 x

化简

1 - 3 x^{2} - 4 x = 0

1 - 3 x^{2} - 4 x = 0

改写成标准形式

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

- 3 x^{2} - 4 x + 1 = 0

使用求根公式求解

- 3 x^{2} - 4 x + 1 = 0

二次方程求根公式:

若

a = - 3, b = - 4, c = 1

x_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 ( - 3 ) \cdot 1}{2 ( - 3 )}

x_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 ( - 3 ) \cdot 1}{2 ( - 3 )}

(- 4)^{2} - 4 (- 3) \cdot 1 = 27

(- 4)^{2} - 4 (- 3) \cdot 1

使用法则

- (- a) = a

= (- 4)^{2} + 4 \cdot 3 \cdot 1

使用指数法则:

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

若

n

是偶数

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

= 4^{2} + 4 \cdot 3 \cdot 1

数字相乘:

4 \cdot 3 \cdot 1 = 12

= 4^{2} + 12

4^{2} = 16

= 16 + 12

数字相加:

16 + 12 = 28

= 28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

x_{1, 2} = \frac{- ( - 4 ) \pm 2 7}{2 ( - 3 )}

将解分隔开

x_{1} = \frac{- ( - 4 ) + 2 7}{2 ( - 3 )}, x_{2} = \frac{- ( - 4 ) - 2 7}{2 ( - 3 )}

x = \frac{- ( - 4 ) + 2 7}{2 ( - 3 )} : - \frac{2 + 7}{3}

\frac{- ( - 4 ) + 2 7}{2 ( - 3 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{4 + 2 7}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{4 + 2 7}{- 6}

使用分式法则:

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

= - \frac{4 + 2 7}{6}

消掉

\frac{4 + 2 7}{6} : \frac{2 + 7}{3}

\frac{4 + 2 7}{6}

分解

4 + 27 : 2 (2 + 7)

4 + 27

改写为

= 2 \cdot 2 + 27

因式分解出通项

2

= 2 (2 + 7)

= \frac{2 ( 2 + 7 )}{6}

约分:

2

= \frac{2 + 7}{3}

= - \frac{2 + 7}{3}

x = \frac{- ( - 4 ) - 2 7}{2 ( - 3 )} : \frac{7 - 2}{3}

\frac{- ( - 4 ) - 2 7}{2 ( - 3 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{4 - 2 7}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{4 - 2 7}{- 6}

使用分式法则:

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

4 - 27 = - (27 - 4)

= \frac{2 7 - 4}{6}

分解

27 - 4 : 2 (7 - 2)

27 - 4

改写为

= 27 - 2 \cdot 2

因式分解出通项

2

= 2 (7 - 2)

= \frac{2 ( 7 - 2 )}{6}

约分:

2

= \frac{7 - 2}{3}

二次方程组的解是:

x = - \frac{2 + 7}{3}, x = \frac{7 - 2}{3}

x = - \frac{2 + 7}{3}, x = \frac{7 - 2}{3}

验证解

找到无定义的点（奇点）:

x = \frac{1}{3}, x = - \frac{1}{3}

取

\frac{3 x + x}{1 - 3 xx}

的分母，令其等于零

解

1 - 3 xx = 0 : x = \frac{1}{3}, x = - \frac{1}{3}

1 - 3 xx = 0

将

1

到右边

1 - 3 xx = 0

两边减去

1

1 - 3 xx - 1 = 0 - 1

化简

- 3 xx = - 1

- 3 xx = - 1

化简

- 3 x^{2} = - 1

两边除以

- 3

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

x^{2} = \frac{1}{3}

对于

x^{2} = f (a)

解为

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

x = \frac{1}{3}, x = - \frac{1}{3}

\frac{1}{3} = \frac{1}{3}

\frac{1}{3}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{3}

使用根式运算法则:

1 = 1

1 = 1

= \frac{1}{3}

- \frac{1}{3} = - \frac{1}{3}

- \frac{1}{3}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{3}

使用根式运算法则:

1 = 1

1 = 1

= - \frac{1}{3}

x = \frac{1}{3}, x = - \frac{1}{3}

以下点无定义

x = \frac{1}{3}, x = - \frac{1}{3}

将不在定义域的点与解相综合:

x = - \frac{2 + 7}{3}, x = \frac{7 - 2}{3}

x = - \frac{2 + 7}{3}, x = \frac{7 - 2}{3}

将解代入原方程进行验证

将它们代入

arctan (3 x) + arctan (x) = \frac{π}{4}

检验解是否符合

去除与方程不符的解。

检验

- \frac{2 + 7}{3}

的解:

假

- \frac{2 + 7}{3}

代入

n = 1

- \frac{2 + 7}{3}

对于

arctan (3 x) + arctan (x) = \frac{π}{4}

代入

x = - \frac{2 + 7}{3}

arctan (3 (- \frac{2 + 7}{3})) + arctan (- \frac{2 + 7}{3}) = \frac{π}{4}

整理后得

- 2.35619 \dots = 0.78539 \dots

\Rightarrow 假

检验

\frac{7 - 2}{3}

的解:

真

\frac{7 - 2}{3}

代入

n = 1

\frac{7 - 2}{3}

对于

arctan (3 x) + arctan (x) = \frac{π}{4}

代入

x = \frac{7 - 2}{3}

arctan (3 \cdot \frac{7 - 2}{3}) + arctan (\frac{7 - 2}{3}) = \frac{π}{4}

整理后得

0.78539 \dots = 0.78539 \dots

\Rightarrow 真

x = \frac{7 - 2}{3}

作图

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