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受欢迎的三角函数 >

cos(3x)-1=sin(3x)

解答

cos (3 x) - 1 = sin (3 x)

解答

x = \frac{π}{2} + \frac{2 πn}{3}, x = \frac{2 π}{3} + \frac{2 πn}{3}

+1

度数

x = 9 0^{\circ} + 12 0^{\circ} n, x = 12 0^{\circ} + 12 0^{\circ} n

求解步骤

cos (3 x) - 1 = sin (3 x)

两边减去

sin (3 x)

cos (3 x) - 1 - sin (3 x) = 0

使用三角恒等式改写

- 1 + cos (3 x) - sin (3 x)

利用以下特性:

sin (x) = cos (\frac{π}{2} - x)

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

使用和差化积恒等式:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

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

2 sin (\frac{3 x + \frac{π}{2} - 3 x}{2}) sin (\frac{3 x - ( \frac{π}{2} - 3 x )}{2}) = 2 sin (\frac{12 x - π}{4})

2 sin (\frac{3 x + \frac{π}{2} - 3 x}{2}) sin (\frac{3 x - ( \frac{π}{2} - 3 x )}{2})

\frac{3 x + \frac{π}{2} - 3 x}{2} = \frac{π}{4}

\frac{3 x + \frac{π}{2} - 3 x}{2}

3 x + \frac{π}{2} - 3 x = \frac{π}{2}

3 x + \frac{π}{2} - 3 x

对同类项分组

= 3 x - 3 x + \frac{π}{2}

同类项相加:

3 x - 3 x = 0

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

使用分式法则:

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

= \frac{π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{π}{4}

= 2 sin (\frac{π}{4}) sin (\frac{3 x - ( - 3 x + \frac{π}{2} )}{2})

\frac{3 x - ( \frac{π}{2} - 3 x )}{2} = \frac{12 x - π}{4}

\frac{3 x - ( \frac{π}{2} - 3 x )}{2}

乘开

3 x - (\frac{π}{2} - 3 x) : 6 x - \frac{π}{2}

3 x - (\frac{π}{2} - 3 x)

- (\frac{π}{2} - 3 x) : - \frac{π}{2} + 3 x

- (\frac{π}{2} - 3 x)

打开括号

= - (\frac{π}{2}) - (- 3 x)

使用加减运算法则

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

= - \frac{π}{2} + 3 x

= 3 x - \frac{π}{2} + 3 x

化简

3 x - \frac{π}{2} + 3 x : 6 x - \frac{π}{2}

3 x - \frac{π}{2} + 3 x

对同类项分组

= 3 x + 3 x - \frac{π}{2}

同类项相加:

3 x + 3 x = 6 x

= 6 x - \frac{π}{2}

= 6 x - \frac{π}{2}

= \frac{6 x - \frac{π}{2}}{2}

化简

6 x - \frac{π}{2} : \frac{12 x - π}{2}

6 x - \frac{π}{2}

将项转换为分式:

6 x = \frac{6 x 2}{2}

= \frac{6 x \cdot 2}{2} - \frac{π}{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{6 x \cdot 2 - π}{2}

数字相乘:

6 \cdot 2 = 12

= \frac{12 x - π}{2}

= \frac{\frac{12 x - π}{2}}{2}

使用分式法则:

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

= \frac{12 x - π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{12 x - π}{4}

= 2 sin (\frac{π}{4}) sin (\frac{12 x - π}{4})

化简

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

使用以下普通恒等式:

sin (\frac{π}{4}) = \frac{2}{2}

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= 2 \cdot \frac{2}{2} sin (\frac{12 x - π}{4})

分式相乘:

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

= \frac{2 \cdot 2 sin ( \frac{12 x - π}{4} )}{2}

约分:

2

= 2 sin (\frac{12 x - π}{4})

= - 1 - 2 sin (\frac{12 x - π}{4})

- 1 - 2 sin (\frac{12 x - π}{4}) = 0

将

1

到右边

- 1 - 2 sin (\frac{12 x - π}{4}) = 0

两边加上

1

- 1 - 2 sin (\frac{12 x - π}{4}) + 1 = 0 + 1

化简

- 2 sin (\frac{12 x - π}{4}) = 1

- 2 sin (\frac{12 x - π}{4}) = 1

两边除以

- 2

- 2 sin (\frac{12 x - π}{4}) = 1

两边除以

- 2

\frac{- 2 sin ( \frac{12 x - π}{4} )}{- 2} = \frac{1}{- 2}

化简

\frac{- 2 sin ( \frac{12 x - π}{4} )}{- 2} = \frac{1}{- 2}

化简

\frac{- 2 sin ( \frac{12 x - π}{4} )}{- 2} : sin (\frac{12 x - π}{4})

\frac{- 2 sin ( \frac{12 x - π}{4} )}{- 2}

使用分式法则:

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

= \frac{2 sin ( \frac{12 x - π}{4} )}{2}

约分:

2

= sin (\frac{12 x - π}{4})

化简

\frac{1}{- 2} : - \frac{2}{2}

\frac{1}{- 2}

使用分式法则:

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

= - \frac{1}{2}

- \frac{1}{2}

有理化:

- \frac{2}{2}

- \frac{1}{2}

乘以共轭根式

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

sin (\frac{12 x - π}{4}) = - \frac{2}{2}

sin (\frac{12 x - π}{4}) = - \frac{2}{2}

sin (\frac{12 x - π}{4}) = - \frac{2}{2}

sin (\frac{12 x - π}{4}) = - \frac{2}{2}

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{12 x - π}{4} = \frac{5 π}{4} + 2 πn, \frac{12 x - π}{4} = \frac{7 π}{4} + 2 πn

\frac{12 x - π}{4} = \frac{5 π}{4} + 2 πn, \frac{12 x - π}{4} = \frac{7 π}{4} + 2 πn

解

\frac{12 x - π}{4} = \frac{5 π}{4} + 2 πn : x = \frac{π}{2} + \frac{2 πn}{3}

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

在两边乘以

4

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

在两边乘以

4

\frac{4 ( 12 x - π )}{4} = 4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn

化简

\frac{4 ( 12 x - π )}{4} = 4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn

化简

\frac{4 ( 12 x - π )}{4} : 12 x - π

\frac{4 ( 12 x - π )}{4}

数字相除:

\frac{4}{4} = 1

= 12 x - π

化简

4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn : 5 π + 8 πn

4 \cdot \frac{5 π}{4} + 4 \cdot 2 πn

4 \cdot \frac{5 π}{4} = 5 π

4 \cdot \frac{5 π}{4}

分式相乘:

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

= \frac{5 π 4}{4}

约分:

4

= 5 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= 5 π + 8 πn

12 x - π = 5 π + 8 πn

12 x - π = 5 π + 8 πn

12 x - π = 5 π + 8 πn

将

π

到右边

12 x - π = 5 π + 8 πn

两边加上

π

12 x - π + π = 5 π + 8 πn + π

化简

12 x = 6 π + 8 πn

12 x = 6 π + 8 πn

两边除以

12

12 x = 6 π + 8 πn

两边除以

12

\frac{12 x}{12} = \frac{6 π}{12} + \frac{8 πn}{12}

化简

\frac{12 x}{12} = \frac{6 π}{12} + \frac{8 πn}{12}

化简

\frac{12 x}{12} : x

\frac{12 x}{12}

数字相除:

\frac{12}{12} = 1

= x

化简

\frac{6 π}{12} + \frac{8 πn}{12} : \frac{π}{2} + \frac{2 πn}{3}

\frac{6 π}{12} + \frac{8 πn}{12}

消掉

\frac{6 π}{12} : \frac{π}{2}

\frac{6 π}{12}

约分:

6

= \frac{π}{2}

= \frac{π}{2} + \frac{8 πn}{12}

消掉

\frac{8 πn}{12} : \frac{2 πn}{3}

\frac{8 πn}{12}

约分:

4

= \frac{2 πn}{3}

= \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}

解

\frac{12 x - π}{4} = \frac{7 π}{4} + 2 πn : x = \frac{2 π}{3} + \frac{2 πn}{3}

\frac{12 x - π}{4} = \frac{7 π}{4} + 2 πn

在两边乘以

4

\frac{12 x - π}{4} = \frac{7 π}{4} + 2 πn

在两边乘以

4

\frac{4 ( 12 x - π )}{4} = 4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn

化简

\frac{4 ( 12 x - π )}{4} = 4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn

化简

\frac{4 ( 12 x - π )}{4} : 12 x - π

\frac{4 ( 12 x - π )}{4}

数字相除:

\frac{4}{4} = 1

= 12 x - π

化简

4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn : 7 π + 8 πn

4 \cdot \frac{7 π}{4} + 4 \cdot 2 πn

4 \cdot \frac{7 π}{4} = 7 π

4 \cdot \frac{7 π}{4}

分式相乘:

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

= \frac{7 π 4}{4}

约分:

4

= 7 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

数字相乘:

4 \cdot 2 = 8

= 8 πn

= 7 π + 8 πn

12 x - π = 7 π + 8 πn

12 x - π = 7 π + 8 πn

12 x - π = 7 π + 8 πn

将

π

到右边

12 x - π = 7 π + 8 πn

两边加上

π

12 x - π + π = 7 π + 8 πn + π

化简

12 x = 8 π + 8 πn

12 x = 8 π + 8 πn

两边除以

12

12 x = 8 π + 8 πn

两边除以

12

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

化简

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

化简

\frac{12 x}{12} : x

\frac{12 x}{12}

数字相除:

\frac{12}{12} = 1

= x

化简

\frac{8 π}{12} + \frac{8 πn}{12} : \frac{2 π}{3} + \frac{2 πn}{3}

\frac{8 π}{12} + \frac{8 πn}{12}

消掉

\frac{8 π}{12} : \frac{2 π}{3}

\frac{8 π}{12}

约分:

4

= \frac{2 π}{3}

= \frac{2 π}{3} + \frac{8 πn}{12}

消掉

\frac{8 πn}{12} : \frac{2 πn}{3}

\frac{8 πn}{12}

约分:

4

= \frac{2 πn}{3}

= \frac{2 π}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} + \frac{2 πn}{3}

x = \frac{2 π}{3} + \frac{2 πn}{3}

x = \frac{π}{2} + \frac{2 πn}{3}, x = \frac{2 π}{3} + \frac{2 πn}{3}

作图

流行的例子

sin(x/3)=(sqrt(2))/2

sin (\frac{x}{3}) = \frac{2}{2}

2cos(3x)=-sqrt(3)

2 cos (3 x) = - 3

cos(x)=cos^2(x)

cos (x) = cos^{2} (x)

- tan (x) = 1

4sin(3x)=7cos(3x)

4 sin (3 x) = 7 cos (3 x)