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受欢迎的三角函数 >

arccot(x-2)=arccot(x-1)+arccot(x)

解答

arccot (x - 2) = arccot (x - 1) + arccot (x)

解答

x \in R 无解

求解步骤

arccot (x - 2) = arccot (x - 1) + arccot (x)

两边减去

arccot (x - 1) + arccot (x)

arccot (x - 2) - arccot (x - 1) - arccot (x) = 0

使用三角恒等式改写

- arccot (- 1 + x) + arccot (- 2 + x) - arccot (x)

使用和差化积恒等式:

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

= - arccot (x) + arccot (\frac{( - 2 + x ) ( - 1 + x ) + 1}{- 1 + x - ( - 2 + x )})

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

\frac{( - 2 + x ) ( - 1 + x ) + 1}{- 1 + x - ( - 2 + x )}

乘开

- 1 + x - (- 2 + x) : 1

- 1 + x - (- 2 + x)

- (- 2 + x) : 2 - x

- (- 2 + x)

打开括号

= - (- 2) - (x)

使用加减运算法则

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

= 2 - x

= - 1 + x + 2 - x

化简

- 1 + x + 2 - x : 1

- 1 + x + 2 - x

对同类项分组

= x - x - 1 + 2

同类项相加:

x - x = 0

= - 1 + 2

数字相加/相减:

- 1 + 2 = 1

= 1

= 1

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

乘开

(- 2 + x) (- 1 + x) + 1 : x^{2} - 3 x + 3

(- 2 + x) (- 1 + x) + 1

乘开

(- 2 + x) (- 1 + x) : 2 - 3 x + x^{2}

(- 2 + x) (- 1 + x)

使用 FOIL 方法:

(a + b) (c + d) = a c + a d + b c + b d

a = - 2, b = x, c = - 1, d = x

= (- 2) (- 1) + (- 2) x + x (- 1) + xx

使用加减运算法则

(- a) (- b) = ab, + (- a) = - a

= 2 \cdot 1 - 2 x - 1 \cdot x + xx

化简

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

2 \cdot 1 - 2 x - 1 \cdot x + xx

同类项相加:

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

= 2 \cdot 1 - 3 x + xx

2 \cdot 1 = 2

2 \cdot 1

数字相乘:

2 \cdot 1 = 2

= 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}

= 2 - 3 x + x^{2}

= 2 - 3 x + x^{2}

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

化简

2 - 3 x + x^{2} + 1 : x^{2} - 3 x + 3

2 - 3 x + x^{2} + 1

对同类项分组

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

数字相加:

2 + 1 = 3

= x^{2} - 3 x + 3

= x^{2} - 3 x + 3

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

使用法则

\frac{a}{1} = a

= x^{2} - 3 x + 3

= - arccot (x) + arccot (x^{2} - 3 x + 3)

使用和差化积恒等式:

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

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

arccot (\frac{( 3 + x ^{2} - 3 x ) x + 1}{x - ( 3 + x ^{2} - 3 x )}) = 0

0 < arccot (x) < π

无解

将解代入原方程进行验证

将它们代入

arccot (x - 2) = arccot (x - 1) + arccot (x)

检验解是否符合

去除与方程不符的解。

x \in R 无解

作图

流行的例子

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