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受欢迎的三角函数 >

arctan(2x)+arctan(3x)= pi/4

解答

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

解答

x = \frac{1}{6}

求解步骤

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

使用三角恒等式改写

arctan (2 x) + arctan (3 x)

使用和差化积恒等式:

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

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

arctan (\frac{2 x + 3 x}{1 - 2 x \cdot 3 x}) = \frac{π}{4}

使用反三角函数性质

arctan (\frac{2 x + 3 x}{1 - 2 x \cdot 3 x}) = \frac{π}{4}

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

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

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

解

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

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

化简

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

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

同类项相加:

2 x + 3 x = 5 x

= \frac{5 x}{1 - 2 \cdot 3 xx}

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

1 - 2 x \cdot 3 x

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

2 x \cdot 3 x

数字相乘:

2 \cdot 3 = 6

= 6 xx

使用指数法则:

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

xx = x^{1 + 1}

= 6 x^{1 + 1}

数字相加:

1 + 1 = 2

= 6 x^{2}

= 1 - 6 x^{2}

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

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

在两边乘以

1 - 6 x^{2}

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

在两边乘以

1 - 6 x^{2}

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

化简

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

化简

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

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

分式相乘:

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

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

约分:

1 - 6 x^{2}

= 5 x

化简

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

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

乘以:

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

= (1 - 6 x^{2})

去除括号:

(a) = a

= 1 - 6 x^{2}

5 x = 1 - 6 x^{2}

5 x = 1 - 6 x^{2}

5 x = 1 - 6 x^{2}

解

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

5 x = 1 - 6 x^{2}

交换两边

1 - 6 x^{2} = 5 x

将

5 x

para o lado esquerdo

1 - 6 x^{2} = 5 x

两边减去

5 x

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

化简

1 - 6 x^{2} - 5 x = 0

1 - 6 x^{2} - 5 x = 0

改写成标准形式

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

- 6 x^{2} - 5 x + 1 = 0

使用求根公式求解

- 6 x^{2} - 5 x + 1 = 0

二次方程求根公式:

若

a = - 6, b = - 5, c = 1

x_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 1}{2 ( - 6 )}

x_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 6 ) \cdot 1}{2 ( - 6 )}

(- 5)^{2} - 4 (- 6) \cdot 1 = 7

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

= 5^{2} + 4 \cdot 6 \cdot 1

数字相乘:

4 \cdot 6 \cdot 1 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数字相加:

25 + 24 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

x_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 ( - 6 )}

将解分隔开

x_{1} = \frac{- ( - 5 ) + 7}{2 ( - 6 )}, x_{2} = \frac{- ( - 5 ) - 7}{2 ( - 6 )}

x = \frac{- ( - 5 ) + 7}{2 ( - 6 )} : - 1

\frac{- ( - 5 ) + 7}{2 ( - 6 )}

去除括号:

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

= \frac{5 + 7}{- 2 \cdot 6}

数字相加:

5 + 7 = 12

= \frac{12}{- 2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{12}{- 12}

使用分式法则:

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

= - \frac{12}{12}

使用法则

\frac{a}{a} = 1

= - 1

x = \frac{- ( - 5 ) - 7}{2 ( - 6 )} : \frac{1}{6}

\frac{- ( - 5 ) - 7}{2 ( - 6 )}

去除括号:

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

= \frac{5 - 7}{- 2 \cdot 6}

数字相减:

5 - 7 = - 2

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

数字相乘:

2 \cdot 6 = 12

= \frac{- 2}{- 12}

使用分式法则:

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

= \frac{2}{12}

约分:

2

= \frac{1}{6}

二次方程组的解是:

x = - 1, x = \frac{1}{6}

x = - 1, x = \frac{1}{6}

验证解

找到无定义的点（奇点）:

x = \frac{1}{6}, x = - \frac{1}{6}

取

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

的分母，令其等于零

解

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

1 - 2 x \cdot 3 x = 0

将

1

到右边

1 - 2 x \cdot 3 x = 0

两边减去

1

1 - 2 x \cdot 3 x - 1 = 0 - 1

化简

- 2 x \cdot 3 x = - 1

- 2 x \cdot 3 x = - 1

化简

- 6 x^{2} = - 1

两边除以

- 6

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

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

对于

x^{2} = f (a)

解为

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

x = \frac{1}{6}, x = - \frac{1}{6}

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

\frac{1}{6}

使用根式运算法则:

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

= \frac{1}{6}

使用根式运算法则:

1 = 1

1 = 1

= \frac{1}{6}

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

- \frac{1}{6}

使用根式运算法则:

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

= - \frac{1}{6}

使用根式运算法则:

1 = 1

1 = 1

= - \frac{1}{6}

x = \frac{1}{6}, x = - \frac{1}{6}

以下点无定义

x = \frac{1}{6}, x = - \frac{1}{6}

将不在定义域的点与解相综合:

x = - 1, x = \frac{1}{6}

x = - 1, x = \frac{1}{6}

将解代入原方程进行验证

将它们代入

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

检验解是否符合

去除与方程不符的解。

检验

- 1

的解:

假

- 1

代入

n = 1

- 1

对于

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

代入

x = - 1

arctan (2 (- 1)) + arctan (3 (- 1)) = \frac{π}{4}

整理后得

- 2.35619 \dots = 0.78539 \dots

\Rightarrow 假

检验

\frac{1}{6}

的解:

真

\frac{1}{6}

代入

n = 1

\frac{1}{6}

对于

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

代入

x = \frac{1}{6}

arctan (2 \cdot \frac{1}{6}) + arctan (3 \cdot \frac{1}{6}) = \frac{π}{4}

整理后得

0.78539 \dots = 0.78539 \dots

\Rightarrow 真

x = \frac{1}{6}

作图

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