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受欢迎的三角函数 >

sin(6x)+sin(x)=0

解答

sin (6 x) + sin (x) = 0

解答

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{7}, x = \frac{2 π}{7} + \frac{4 πn}{7}

+1

度数

x = 3 6^{\circ} + 14 4^{\circ} n, x = 10 8^{\circ} + 14 4^{\circ} n, x = 0^{\circ} + 102.85714 \dots^{\circ} n, x = 51.42857 \dots^{\circ} + 102.85714 \dots^{\circ} n

求解步骤

sin (6 x) + sin (x) = 0

使用三角恒等式改写

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

化简

2 sin (\frac{6 x + x}{2}) cos (\frac{6 x - x}{2}) : 2 sin (\frac{7 x}{2}) cos (\frac{5 x}{2})

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

同类项相加:

6 x + x = 7 x

= 2 sin (\frac{7 x}{2}) cos (\frac{6 x - x}{2})

同类项相加:

6 x - x = 5 x

= 2 sin (\frac{7 x}{2}) cos (\frac{5 x}{2})

= 2 sin (\frac{7 x}{2}) cos (\frac{5 x}{2})

2 cos (\frac{5 x}{2}) sin (\frac{7 x}{2}) = 0

分别求解每个部分

cos (\frac{5 x}{2}) = 0 or sin (\frac{7 x}{2}) = 0

cos (\frac{5 x}{2}) = 0 : x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

cos (\frac{5 x}{2}) = 0

cos (\frac{5 x}{2}) = 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}

\frac{5 x}{2} = \frac{π}{2} + 2 πn, \frac{5 x}{2} = \frac{3 π}{2} + 2 πn

\frac{5 x}{2} = \frac{π}{2} + 2 πn, \frac{5 x}{2} = \frac{3 π}{2} + 2 πn

解

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

\frac{5 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{5 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

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

化简

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

化简

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

数字相乘:

2 \cdot 5 = 10

= \frac{10 x}{2}

数字相除:

\frac{10}{2} = 5

= 5 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

5 x = π + 4 πn

5 x = π + 4 πn

5 x = π + 4 πn

两边除以

5

5 x = π + 4 πn

两边除以

5

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5}

化简

x = \frac{π}{5} + \frac{4 πn}{5}

x = \frac{π}{5} + \frac{4 πn}{5}

解

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

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{5 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{2 \cdot 5 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 5 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 5 x}{2} : 5 x

\frac{2 \cdot 5 x}{2}

数字相乘:

2 \cdot 5 = 10

= \frac{10 x}{2}

数字相除:

\frac{10}{2} = 5

= 5 x

化简

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

分式相乘:

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

= \frac{3 π 2}{2}

约分:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

5 x = 3 π + 4 πn

两边除以

5

5 x = 3 π + 4 πn

两边除以

5

\frac{5 x}{5} = \frac{3 π}{5} + \frac{4 πn}{5}

化简

x = \frac{3 π}{5} + \frac{4 πn}{5}

x = \frac{3 π}{5} + \frac{4 πn}{5}

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

sin (\frac{7 x}{2}) = 0 : x = \frac{4 πn}{7}, x = \frac{2 π}{7} + \frac{4 πn}{7}

sin (\frac{7 x}{2}) = 0

sin (\frac{7 x}{2}) = 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}

\frac{7 x}{2} = 0 + 2 πn, \frac{7 x}{2} = π + 2 πn

\frac{7 x}{2} = 0 + 2 πn, \frac{7 x}{2} = π + 2 πn

解

\frac{7 x}{2} = 0 + 2 πn : x = \frac{4 πn}{7}

\frac{7 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{7 x}{2} = 2 πn

在两边乘以

2

\frac{7 x}{2} = 2 πn

在两边乘以

2

\frac{2 \cdot 7 x}{2} = 2 \cdot 2 πn

化简

7 x = 4 πn

7 x = 4 πn

两边除以

7

7 x = 4 πn

两边除以

7

\frac{7 x}{7} = \frac{4 πn}{7}

化简

x = \frac{4 πn}{7}

x = \frac{4 πn}{7}

解

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

\frac{7 x}{2} = π + 2 πn

在两边乘以

2

\frac{7 x}{2} = π + 2 πn

在两边乘以

2

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

化简

7 x = 2 π + 4 πn

7 x = 2 π + 4 πn

两边除以

7

7 x = 2 π + 4 πn

两边除以

7

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

化简

x = \frac{2 π}{7} + \frac{4 πn}{7}

x = \frac{2 π}{7} + \frac{4 πn}{7}

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

合并所有解

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{7}, x = \frac{2 π}{7} + \frac{4 πn}{7}

作图

流行的例子

30arcsin(x)=5pi

30 arcsin (x) = 5 π

3(tan(x))+sqrt(12)=sqrt(3)

3 (tan (x)) + 12 = 3

tan(3x)-sqrt(3)=0

tan (3 x) - 3 = 0

3sin(x)-2=5sin(x)-1

3 sin (x) - 2 = 5 sin (x) - 1

1+cos(θ)=3cos(θ)

1 + cos (θ) = 3 cos (θ)