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受欢迎的三角函数 >

arcsin(x)-arccos(x)=arcsin(3x-2)

解答

arcsin (x) - arccos (x) = arcsin (3 x - 2)

解答

x = 1, x = \frac{1}{2}

求解步骤

arcsin (x) - arccos (x) = arcsin (3 x - 2)

a = b \Rightarrow sin (a) = sin (b)

sin (arcsin (x) - arccos (x)) = sin (arcsin (3 x - 2))

利用以下特性:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

sin (arcsin (x)) cos (arccos (x)) - cos (arcsin (x)) sin (arccos (x)) = sin (arcsin (3 x - 2))

利用以下特性:

sin (arcsin (x)) = x

利用以下特性:

cos (arccos (x)) = x

利用以下特性:

cos (arcsin (x)) = 1 - x^{2}

利用以下特性:

sin (arccos (x)) = 1 - x^{2}

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

解

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

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

展开

xx - 1 - x^{2} 1 - x^{2} : 2 x^{2} - 1

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

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

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

使用根式运算法则:

a a = a

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

= 1 - x^{2}

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

展开

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

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

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

- (1 - x^{2})

打开括号

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

使用加减运算法则

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

= - 1 + x^{2}

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

化简

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

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

对同类项分组

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

同类项相加:

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

= 2 x^{2} - 1

= 2 x^{2} - 1

= 2 x^{2} - 1

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

解

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

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

将

2

para o lado esquerdo

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

两边加上

2

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

化简

2 x^{2} + 1 = 3 x

2 x^{2} + 1 = 3 x

将

3 x

para o lado esquerdo

2 x^{2} + 1 = 3 x

两边减去

3 x

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

化简

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 2, b = - 3, c = 1

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

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

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

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

使用指数法则:

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

若

n

是偶数

(- 3)^{2} = 3^{2}

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

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数字相减:

9 - 8 = 1

= 1

使用法则

1 = 1

= 1

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相加:

3 + 1 = 4

= \frac{4}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{4}{4}

使用法则

\frac{a}{a} = 1

= 1

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

\frac{- ( - 3 ) - 1}{2 \cdot 2}

使用法则

- (- a) = a

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

数字相减:

3 - 1 = 2

= \frac{2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

约分:

2

= \frac{1}{2}

二次方程组的解是:

x = 1, x = \frac{1}{2}

x = 1, x = \frac{1}{2}

验证解:

x = 1

真

, x = \frac{1}{2}

真

将它们代入

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

检验解是否符合

去除与方程不符的解。

代入

x = 1 :

真

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

1 \cdot 1 - 1 - 1^{2} 1 - 1^{2} = 1

1 \cdot 1 - 1 - 1^{2} 1 - 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

= 1 \cdot 1 - 1 - 1 1 - 1

1 \cdot 1 = 1

1 \cdot 1

数字相乘:

1 \cdot 1 = 1

= 1

1 - 1 1 - 1 = 0

1 - 1 1 - 1

1 - 1 = 0

1 - 1

数字相减:

1 - 1 = 0

= 0

使用法则

0 = 0

= 0

= 0 \cdot 1 - 1

1 - 1 = 0

1 - 1

数字相减:

1 - 1 = 0

= 0

使用法则

0 = 0

= 0

= 0 \cdot 0

数字相乘:

0 \cdot 0 = 0

= 0

= 1 - 0

数字相减:

1 - 0 = 1

= 1

3 \cdot 1 - 2 = 1

3 \cdot 1 - 2

数字相乘:

3 \cdot 1 = 3

= 3 - 2

数字相减:

3 - 2 = 1

= 1

1 = 1

真

代入

x = \frac{1}{2} :

真

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

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

(\frac{1}{2}) (\frac{1}{2}) - 1 - (\frac{1}{2})^{2} 1 - (\frac{1}{2})^{2}

去除括号:

(a) = a

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

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

分式相乘:

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

= \frac{1 \cdot 1}{2 \cdot 2}

数字相乘:

1 \cdot 1 = 1

= \frac{1}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{1}{4}

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

1 - (\frac{1}{2})^{2} 1 - (\frac{1}{2})^{2}

使用根式运算法则:

a a = a

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

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

(\frac{1}{2})^{2} = \frac{1}{4}

(\frac{1}{2})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

使用法则

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

将项转换为分式:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{1}{4}

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 4 - 1}{4}

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

数字相乘:

1 \cdot 4 = 4

= 4 - 1

数字相减:

4 - 1 = 3

= 3

= \frac{3}{4}

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

使用法则

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

= \frac{1 - 3}{4}

数字相减:

1 - 3 = - 2

= \frac{- 2}{4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

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

3 (\frac{1}{2}) - 2

去除括号:

(a) = a

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

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

3 \cdot \frac{1}{2}

分式相乘:

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

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

数字相乘:

1 \cdot 3 = 3

= \frac{3}{2}

= \frac{3}{2} - 2

将项转换为分式:

2 = \frac{2 \cdot 2}{2}

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

- 2 \cdot 2 + 3 = - 1

- 2 \cdot 2 + 3

数字相乘:

2 \cdot 2 = 4

= - 4 + 3

数字相加/相减:

- 4 + 3 = - 1

= - 1

= \frac{- 1}{2}

使用分式法则:

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

= - \frac{1}{2}

- \frac{1}{2} = - \frac{1}{2}

真

解为

x = 1, x = \frac{1}{2}

x = 1, x = \frac{1}{2}

将解代入原方程进行验证

将它们代入

arcsin (x) - arccos (x) = arcsin (3 x - 2)

检验解是否符合

去除与方程不符的解。

检验

1

的解:

真

1

代入

n = 1

1

对于

arcsin (x) - arccos (x) = arcsin (3 x - 2)

代入

x = 1

arcsin (1) - arccos (1) = arcsin (3 \cdot 1 - 2)

整理后得

1.57079 \dots = 1.57079 \dots

\Rightarrow 真

检验

\frac{1}{2}

的解:

真

\frac{1}{2}

代入

n = 1

\frac{1}{2}

对于

arcsin (x) - arccos (x) = arcsin (3 x - 2)

代入

x = \frac{1}{2}

arcsin (\frac{1}{2}) - arccos (\frac{1}{2}) = arcsin (3 \cdot \frac{1}{2} - 2)

整理后得

- 0.52359 \dots = - 0.52359 \dots

\Rightarrow 真

x = 1, x = \frac{1}{2}

作图

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