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受欢迎的三角函数 >

sin(4x)=cos(x/2)

解答

sin (4 x) = cos (\frac{x}{2})

解答

x \in R 无解

求解步骤

sin (4 x) = cos (\frac{x}{2})

使用三角恒等式改写

sin (4 x) = cos (\frac{x}{2})

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

sin (4 x) = sin (\frac{π}{2} - \frac{x}{2})

sin (4 x) = sin (\frac{π}{2} - \frac{x}{2})

使用反三角函数性质

sin (4 x) = sin (\frac{π}{2} - \frac{x}{2})

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn, 4 x = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn, 4 x = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn : x = \frac{π + 4 πn}{9}

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn

将

\frac{x}{2}

para o lado esquerdo

4 x = \frac{π}{2} - \frac{x}{2} + 2 πn

两边加上

\frac{x}{2}

4 x + \frac{x}{2} = \frac{π}{2} - \frac{x}{2} + 2 πn + \frac{x}{2}

化简

4 x + \frac{x}{2} = \frac{π}{2} - \frac{x}{2} + 2 πn + \frac{x}{2}

化简

4 x + \frac{x}{2} : \frac{9 x}{2}

4 x + \frac{x}{2}

将项转换为分式:

4 x = \frac{4 x 2}{2}

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

x + 4 x \cdot 2 = 9 x

x + 4 x \cdot 2

数字相乘:

4 \cdot 2 = 8

= x + 8 x

同类项相加:

x + 8 x = 9 x

= 9 x

= \frac{9 x}{2}

化简

\frac{π}{2} - \frac{x}{2} + 2 πn + \frac{x}{2} : \frac{π}{2} + 2 πn

\frac{π}{2} - \frac{x}{2} + 2 πn + \frac{x}{2}

同类项相加:

- \frac{x}{2} + \frac{x}{2} = 0

= \frac{π}{2} + 2 πn

\frac{9 x}{2} = \frac{π}{2} + 2 πn

\frac{9 x}{2} = \frac{π}{2} + 2 πn

\frac{9 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{9 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{2 \cdot 9 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 9 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 9 x}{2} : 9 x

\frac{2 \cdot 9 x}{2}

数字相乘:

2 \cdot 9 = 18

= \frac{18 x}{2}

数字相除:

\frac{18}{2} = 9

= 9 x

化简

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

分式相乘:

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

= \frac{π 2}{2}

约分:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

9 x = π + 4 πn

9 x = π + 4 πn

9 x = π + 4 πn

两边除以

9

9 x = π + 4 πn

两边除以

9

\frac{9 x}{9} = \frac{π}{9} + \frac{4 πn}{9}

化简

\frac{9 x}{9} = \frac{π}{9} + \frac{4 πn}{9}

化简

\frac{9 x}{9} : x

\frac{9 x}{9}

数字相除:

\frac{9}{9} = 1

= x

化简

\frac{π}{9} + \frac{4 πn}{9} : \frac{π + 4 πn}{9}

\frac{π}{9} + \frac{4 πn}{9}

使用法则

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

= \frac{π + 4 πn}{9}

x = \frac{π + 4 πn}{9}

x = \frac{π + 4 πn}{9}

x = \frac{π + 4 πn}{9}

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

4 x = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn

将

(\frac{π}{2} - \frac{x}{2})

para o lado esquerdo

4 x = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn

两边加上

(\frac{π}{2} - \frac{x}{2})

4 x + \frac{π}{2} - \frac{x}{2} = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn + \frac{π}{2} - \frac{x}{2}

化简

4 x + \frac{π}{2} - \frac{x}{2} = π - (\frac{π}{2} - \frac{x}{2}) + 2 πn + \frac{π}{2} - \frac{x}{2}

化简

4 x + \frac{π}{2} - \frac{x}{2} : \frac{π + 7 x}{2}

4 x + \frac{π}{2} - \frac{x}{2}

合并分式

\frac{π}{2} - \frac{x}{2} : \frac{π - x}{2}

使用法则

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

= \frac{π - x}{2}

= 4 x + \frac{- x + π}{2}

将项转换为分式:

4 x = \frac{4 x 2}{2}

= \frac{π - x}{2} + \frac{4 x \cdot 2}{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{π - x + 4 x \cdot 2}{2}

π - x + 4 x \cdot 2 = π + 7 x

π - x + 4 x \cdot 2

数字相乘:

4 \cdot 2 = 8

= π - x + 8 x

同类项相加:

- x + 8 x = 7 x

= π + 7 x

= \frac{π + 7 x}{2}

化简

π - (\frac{π}{2} - \frac{x}{2}) + 2 πn + \frac{π}{2} - \frac{x}{2} : π + 2 πn

π - (\frac{π}{2} - \frac{x}{2}) + 2 πn + \frac{π}{2} - \frac{x}{2}

同类项相加:

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

= π + 2 πn

\frac{π + 7 x}{2} = π + 2 πn

\frac{π + 7 x}{2} = π + 2 πn

\frac{π + 7 x}{2} = π + 2 πn

在两边乘以

2

\frac{π + 7 x}{2} = π + 2 πn

在两边乘以

2

\frac{2 ( π + 7 x )}{2} = 2 π + 2 \cdot 2 πn

化简

π + 7 x = 2 π + 4 πn

π + 7 x = 2 π + 4 πn

将

π

到右边

π + 7 x = 2 π + 4 πn

两边减去

π

π + 7 x - π = 2 π + 4 πn - π

化简

7 x = π + 4 πn

7 x = π + 4 πn

两边除以

7

7 x = π + 4 πn

两边除以

7

\frac{7 x}{7} = \frac{π}{7} + \frac{4 πn}{7}

化简

\frac{7 x}{7} = \frac{π}{7} + \frac{4 πn}{7}

化简

\frac{7 x}{7} : x

\frac{7 x}{7}

数字相除:

\frac{7}{7} = 1

= x

化简

\frac{π}{7} + \frac{4 πn}{7} : \frac{π + 4 πn}{7}

\frac{π}{7} + \frac{4 πn}{7}

使用法则

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

= \frac{π + 4 πn}{7}

x = \frac{π + 4 πn}{7}

x = \frac{π + 4 πn}{7}

x = \frac{π + 4 πn}{7}

因为方程对以下值无定义:

\frac{π + 4 πn}{9}, \frac{π + 4 πn}{7}

x \in R 无解

作图

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