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受欢迎的三角函数 >

sin(5x)=cos(x)

解答

sin (5 x) = cos (x)

解答

x = \frac{π + 4 πn}{12}, x = \frac{π + 4 πn}{8}

+1

度数

x = 1 5^{\circ} + 6 0^{\circ} n, x = 22. 5^{\circ} + 9 0^{\circ} n

求解步骤

sin (5 x) = cos (x)

使用三角恒等式改写

sin (5 x) = cos (x)

利用以下特性:

cos (x) = sin (\frac{π}{2} - x)

sin (5 x) = sin (\frac{π}{2} - x)

sin (5 x) = sin (\frac{π}{2} - x)

使用反三角函数性质

sin (5 x) = sin (\frac{π}{2} - x)

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

5 x = \frac{π}{2} - x + 2 πn, 5 x = π - (\frac{π}{2} - x) + 2 πn

5 x = \frac{π}{2} - x + 2 πn, 5 x = π - (\frac{π}{2} - x) + 2 πn

5 x = \frac{π}{2} - x + 2 πn : x = \frac{π + 4 πn}{12}

5 x = \frac{π}{2} - x + 2 πn

将

x

para o lado esquerdo

5 x = \frac{π}{2} - x + 2 πn

两边加上

x

5 x + x = \frac{π}{2} - x + 2 πn + x

化简

6 x = \frac{π}{2} + 2 πn

6 x = \frac{π}{2} + 2 πn

两边除以

6

6 x = \frac{π}{2} + 2 πn

两边除以

6

\frac{6 x}{6} = \frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} = \frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} : x

\frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{π}{2}}{6} + \frac{2 πn}{6} : \frac{π + 4 πn}{12}

\frac{\frac{π}{2}}{6} + \frac{2 πn}{6}

使用法则

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

= \frac{\frac{π}{2} + 2 πn}{6}

化简

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

\frac{π}{2} + 2 πn

将项转换为分式:

2 πn = \frac{2 πn 2}{2}

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

数字相乘:

2 \cdot 2 = 4

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{6}

使用分式法则:

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

= \frac{π + 4 πn}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{π + 4 πn}{12}

x = \frac{π + 4 πn}{12}

x = \frac{π + 4 πn}{12}

x = \frac{π + 4 πn}{12}

5 x = π - (\frac{π}{2} - x) + 2 πn : x = \frac{π + 4 πn}{8}

5 x = π - (\frac{π}{2} - x) + 2 πn

展开

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

π - (\frac{π}{2} - x) + 2 πn

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

- (\frac{π}{2} - x)

打开括号

= - (\frac{π}{2}) - (- x)

使用加减运算法则

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

= - \frac{π}{2} + x

= π - \frac{π}{2} + x + 2 πn

5 x = π - \frac{π}{2} + x + 2 πn

将

x

para o lado esquerdo

5 x = π - \frac{π}{2} + x + 2 πn

两边减去

x

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

化简

4 x = π - \frac{π}{2} + 2 πn

4 x = π - \frac{π}{2} + 2 πn

两边除以

4

4 x = π - \frac{π}{2} + 2 πn

两边除以

4

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

化简

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

化简

\frac{4 x}{4} : x

\frac{4 x}{4}

数字相除:

\frac{4}{4} = 1

= x

化简

\frac{π}{4} - \frac{\frac{π}{2}}{4} + \frac{2 πn}{4} : \frac{π + 4 πn}{8}

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

使用法则

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

= \frac{π - \frac{π}{2} + 2 πn}{4}

化简

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

π - \frac{π}{2} + 2 πn

将项转换为分式:

π = \frac{π 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 2}{2} - \frac{π}{2} + \frac{2 πn \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 2 - π + 2 πn \cdot 2}{2}

π 2 - π + 2 πn \cdot 2 = π + 4 πn

π 2 - π + 2 πn \cdot 2

同类项相加:

2 π - π = π

= π + 2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= π + 4 πn

= \frac{π + 4 πn}{2}

= \frac{\frac{π + 4 πn}{2}}{4}

使用分式法则:

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

= \frac{π + 4 πn}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{12}, x = \frac{π + 4 πn}{8}

x = \frac{π + 4 πn}{12}, x = \frac{π + 4 πn}{8}

作图

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