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受欢迎的三角函数 >

solvefor x,sin(x/x)=0.7

解答

求解

x, sin (\frac{x}{x}) = 0.7

解答

x \in R 无解

求解步骤

sin (\frac{x}{x}) = 0.7

使用反三角函数性质

sin (\frac{x}{x}) = 0.7

sin (\frac{x}{x}) = 0.7

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

\frac{x}{x} = arcsin (0.7) + 2 πn, \frac{x}{x} = π - arcsin (0.7) + 2 πn

\frac{x}{x} = arcsin (0.7) + 2 πn, \frac{x}{x} = π - arcsin (0.7) + 2 πn

解

\frac{x}{x} = arcsin (0.7) + 2 πn : n = \frac{1 - arcsin ( 0.7 )}{2 π}

\frac{x}{x} = arcsin (0.7) + 2 πn

化简

\frac{x}{x} : 1

\frac{x}{x}

使用法则

\frac{a}{a} = 1

= 1

1 = arcsin (0.7) + 2 πn

交换两边

arcsin (0.7) + 2 πn = 1

将

arcsin (0.7)

到右边

arcsin (0.7) + 2 πn = 1

两边减去

arcsin (0.7)

arcsin (0.7) + 2 πn - arcsin (0.7) = 1 - arcsin (0.7)

化简

2 πn = 1 - arcsin (0.7)

2 πn = 1 - arcsin (0.7)

两边除以

2 π

2 πn = 1 - arcsin (0.7)

两边除以

2 π

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π}

化简

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π}

化简

\frac{2 πn}{2 π} : n

\frac{2 πn}{2 π}

数字相除:

\frac{2}{2} = 1

= \frac{πn}{π}

约分:

π

= n

化简

\frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π} : \frac{1 - arcsin ( 0.7 )}{2 π}

\frac{1}{2 π} - \frac{arcsin ( 0.7 )}{2 π}

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 - arcsin ( 0.7 )}{2 π}

n = \frac{1 - arcsin ( 0.7 )}{2 π}

n = \frac{1 - arcsin ( 0.7 )}{2 π}

n = \frac{1 - arcsin ( 0.7 )}{2 π}

验证解

找到无定义的点（奇点）:

x = 0

取

\frac{x}{x}

的分母，令其等于零

x = 0

以下点无定义

x = 0

将不在定义域的点与解相综合:

n = \frac{1 - arcsin ( 0.7 )}{2 π}

解

\frac{x}{x} = π - arcsin (0.7) + 2 πn : n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

\frac{x}{x} = π - arcsin (0.7) + 2 πn

化简

\frac{x}{x} : 1

\frac{x}{x}

使用法则

\frac{a}{a} = 1

= 1

1 = π - arcsin (0.7) + 2 πn

交换两边

π - arcsin (0.7) + 2 πn = 1

将

π

到右边

π - arcsin (0.7) + 2 πn = 1

两边减去

π

π - arcsin (0.7) + 2 πn - π = 1 - π

化简

- arcsin (0.7) + 2 πn = 1 - π

- arcsin (0.7) + 2 πn = 1 - π

将

arcsin (0.7)

到右边

- arcsin (0.7) + 2 πn = 1 - π

两边加上

arcsin (0.7)

- arcsin (0.7) + 2 πn + arcsin (0.7) = 1 - π + arcsin (0.7)

化简

2 πn = 1 - π + arcsin (0.7)

2 πn = 1 - π + arcsin (0.7)

两边除以

2 π

2 πn = 1 - π + arcsin (0.7)

两边除以

2 π

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

化简

\frac{2 πn}{2 π} = \frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

化简

\frac{2 πn}{2 π} : n

\frac{2 πn}{2 π}

数字相除:

\frac{2}{2} = 1

= \frac{πn}{π}

约分:

π

= n

化简

\frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π} : - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

\frac{1}{2 π} - \frac{π}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

消掉

\frac{π}{2 π} : \frac{1}{2}

\frac{π}{2 π}

约分:

π

= \frac{1}{2}

= \frac{1}{2 π} - \frac{1}{2} + \frac{arcsin ( 0.7 )}{2 π}

合并分式

\frac{1}{2 π} + \frac{arcsin ( 0.7 )}{2 π} : \frac{1 + arcsin ( 0.7 )}{2 π}

\frac{1}{2 π} + \frac{arcsin ( 0.7 )}{2 π}

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 + arcsin ( 0.7 )}{2 π}

= - \frac{1}{2} + \frac{arcsin ( 0.7 ) + 1}{2 π}

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

验证解

找到无定义的点（奇点）:

x = 0

取

\frac{x}{x}

的分母，令其等于零

x = 0

以下点无定义

x = 0

将不在定义域的点与解相综合:

n = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

x = \frac{1 - arcsin ( 0.7 )}{2 π}, x = - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

因为方程对以下值无定义:

\frac{1 - arcsin ( 0.7 )}{2 π}, - \frac{1}{2} + \frac{1 + arcsin ( 0.7 )}{2 π}

x \in R 无解

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