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受欢迎的三角函数 >

sin(6x)-sin(2x)=0

解答

sin (6 x) - sin (2 x) = 0

解答

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}, x = πn, x = \frac{π}{2} + πn

+1

度数

x = 22. 5^{\circ} + 9 0^{\circ} n, x = 67. 5^{\circ} + 9 0^{\circ} n, x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n

求解步骤

sin (6 x) - sin (2 x) = 0

使用三角恒等式改写

- sin (2 x) + sin (6 x)

使用和差化积恒等式:

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

= 2 sin (\frac{6 x - 2 x}{2}) cos (\frac{6 x + 2 x}{2})

化简

2 sin (\frac{6 x - 2 x}{2}) cos (\frac{6 x + 2 x}{2}) : 2 sin (2 x) cos (4 x)

2 sin (\frac{6 x - 2 x}{2}) cos (\frac{6 x + 2 x}{2})

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

\frac{6 x - 2 x}{2}

同类项相加:

6 x - 2 x = 4 x

= \frac{4 x}{2}

数字相除:

\frac{4}{2} = 2

= 2 x

= 2 sin (2 x) cos (\frac{6 x + 2 x}{2})

\frac{6 x + 2 x}{2} = 4 x

\frac{6 x + 2 x}{2}

同类项相加:

6 x + 2 x = 8 x

= \frac{8 x}{2}

数字相除:

\frac{8}{2} = 4

= 4 x

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

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

2 cos (4 x) sin (2 x) = 0

分别求解每个部分

cos (4 x) = 0 or sin (2 x) = 0

cos (4 x) = 0 : x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}

cos (4 x) = 0

cos (4 x) = 0

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

4 x = \frac{π}{2} + 2 πn, 4 x = \frac{3 π}{2} + 2 πn

4 x = \frac{π}{2} + 2 πn, 4 x = \frac{3 π}{2} + 2 πn

解

4 x = \frac{π}{2} + 2 πn : x = \frac{π}{8} + \frac{πn}{2}

4 x = \frac{π}{2} + 2 πn

两边除以

4

4 x = \frac{π}{2} + 2 πn

两边除以

4

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

化简

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

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

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

使用分式法则:

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

= \frac{π}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

= \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

解

4 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{8} + \frac{πn}{2}

4 x = \frac{3 π}{2} + 2 πn

两边除以

4

4 x = \frac{3 π}{2} + 2 πn

两边除以

4

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

化简

\frac{4 x}{4} = \frac{\frac{3 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{\frac{3 π}{2}}{4} + \frac{2 πn}{4} : \frac{3 π}{8} + \frac{πn}{2}

\frac{\frac{3 π}{2}}{4} + \frac{2 πn}{4}

\frac{\frac{3 π}{2}}{4} = \frac{3 π}{8}

\frac{\frac{3 π}{2}}{4}

使用分式法则:

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

= \frac{3 π}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{3 π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

约分:

2

= \frac{πn}{2}

= \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}

sin (2 x) = 0 : x = πn, x = \frac{π}{2} + πn

sin (2 x) = 0

sin (2 x) = 0

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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

解

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

两边除以

2

2 x = 2 πn

两边除以

2

\frac{2 x}{2} = \frac{2 πn}{2}

化简

x = πn

x = πn

解

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

两边除以

2

2 x = π + 2 πn

两边除以

2

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

化简

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

合并所有解

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}, x = πn, x = \frac{π}{2} + πn

作图

流行的例子

solvefor z,cos(x)+cos(y)+cos(z)=0

so l v e f or z, cos (x) + cos (y) + cos (z) = 0

12sin^2(x)+cos(x)-6=0

12 sin^{2} (x) + cos (x) - 6 = 0

2 sin (2 θ) - 1 = 0

cos (\frac{1}{x}) = 0

solvefor y,x=sinh(y)

so l v e f or y, x = sinh (y)