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受欢迎的三角函数 >

sin(3θ)-sin(6θ)=0

解答

sin (3 θ) - sin (6 θ) = 0

解答

θ = \frac{π}{9} + \frac{4 πn}{9}, θ = \frac{π}{3} + \frac{4 πn}{9}, θ = \frac{4 πn}{3}, θ = \frac{2 π}{3} + \frac{4 πn}{3}

+1

度数

θ = 2 0^{\circ} + 8 0^{\circ} n, θ = 6 0^{\circ} + 8 0^{\circ} n, θ = 0^{\circ} + 24 0^{\circ} n, θ = 12 0^{\circ} + 24 0^{\circ} n

求解步骤

sin (3 θ) - sin (6 θ) = 0

使用三角恒等式改写

sin (3 θ) - sin (6 θ)

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{3 θ - 6 θ}{2}) cos (\frac{3 θ + 6 θ}{2})

化简

2 sin (\frac{3 θ - 6 θ}{2}) cos (\frac{3 θ + 6 θ}{2}) : - 2 cos (\frac{9 θ}{2}) sin (\frac{3 θ}{2})

2 sin (\frac{3 θ - 6 θ}{2}) cos (\frac{3 θ + 6 θ}{2})

\frac{3 θ - 6 θ}{2} = - \frac{3 θ}{2}

\frac{3 θ - 6 θ}{2}

同类项相加:

3 θ - 6 θ = - 3 θ

= \frac{- 3 θ}{2}

使用分式法则:

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

= - \frac{3 θ}{2}

= 2 sin (- \frac{3 θ}{2}) cos (\frac{3 θ + 6 θ}{2})

使用负角恒等式:

sin (- x) = - sin (x)

= 2 cos (\frac{3 θ + 6 θ}{2}) (- sin (\frac{3 θ}{2}))

去除括号:

(- a) = - a

= - 2 cos (\frac{3 θ + 6 θ}{2}) sin (\frac{3 θ}{2})

同类项相加:

3 θ + 6 θ = 9 θ

= - 2 cos (\frac{9 θ}{2}) sin (\frac{3 θ}{2})

= - 2 cos (\frac{9 θ}{2}) sin (\frac{3 θ}{2})

- 2 cos (\frac{9 θ}{2}) sin (\frac{3 θ}{2}) = 0

分别求解每个部分

cos (\frac{9 θ}{2}) = 0 or sin (\frac{3 θ}{2}) = 0

cos (\frac{9 θ}{2}) = 0 : θ = \frac{π}{9} + \frac{4 πn}{9}, θ = \frac{π}{3} + \frac{4 πn}{9}

cos (\frac{9 θ}{2}) = 0

cos (\frac{9 θ}{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{9 θ}{2} = \frac{π}{2} + 2 πn, \frac{9 θ}{2} = \frac{3 π}{2} + 2 πn

\frac{9 θ}{2} = \frac{π}{2} + 2 πn, \frac{9 θ}{2} = \frac{3 π}{2} + 2 πn

解

\frac{9 θ}{2} = \frac{π}{2} + 2 πn : θ = \frac{π}{9} + \frac{4 πn}{9}

\frac{9 θ}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{9 θ}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{2 \cdot 9 θ}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 9 θ}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 9 θ}{2} : 9 θ

\frac{2 \cdot 9 θ}{2}

数字相乘:

2 \cdot 9 = 18

= \frac{18 θ}{2}

数字相除:

\frac{18}{2} = 9

= 9 θ

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

9 θ = π + 4 πn

9 θ = π + 4 πn

9 θ = π + 4 πn

两边除以

9

9 θ = π + 4 πn

两边除以

9

\frac{9 θ}{9} = \frac{π}{9} + \frac{4 πn}{9}

化简

θ = \frac{π}{9} + \frac{4 πn}{9}

θ = \frac{π}{9} + \frac{4 πn}{9}

解

\frac{9 θ}{2} = \frac{3 π}{2} + 2 πn : θ = \frac{π}{3} + \frac{4 πn}{9}

\frac{9 θ}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{9 θ}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{2 \cdot 9 θ}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 9 θ}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 9 θ}{2} : 9 θ

\frac{2 \cdot 9 θ}{2}

数字相乘:

2 \cdot 9 = 18

= \frac{18 θ}{2}

数字相除:

\frac{18}{2} = 9

= 9 θ

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

9 θ = 3 π + 4 πn

9 θ = 3 π + 4 πn

9 θ = 3 π + 4 πn

两边除以

9

9 θ = 3 π + 4 πn

两边除以

9

\frac{9 θ}{9} = \frac{3 π}{9} + \frac{4 πn}{9}

化简

θ = \frac{π}{3} + \frac{4 πn}{9}

θ = \frac{π}{3} + \frac{4 πn}{9}

θ = \frac{π}{9} + \frac{4 πn}{9}, θ = \frac{π}{3} + \frac{4 πn}{9}

sin (\frac{3 θ}{2}) = 0 : θ = \frac{4 πn}{3}, θ = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{3 θ}{2}) = 0

sin (\frac{3 θ}{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{3 θ}{2} = 0 + 2 πn, \frac{3 θ}{2} = π + 2 πn

\frac{3 θ}{2} = 0 + 2 πn, \frac{3 θ}{2} = π + 2 πn

解

\frac{3 θ}{2} = 0 + 2 πn : θ = \frac{4 πn}{3}

\frac{3 θ}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 θ}{2} = 2 πn

在两边乘以

2

\frac{3 θ}{2} = 2 πn

在两边乘以

2

\frac{2 \cdot 3 θ}{2} = 2 \cdot 2 πn

化简

3 θ = 4 πn

3 θ = 4 πn

两边除以

3

3 θ = 4 πn

两边除以

3

\frac{3 θ}{3} = \frac{4 πn}{3}

化简

θ = \frac{4 πn}{3}

θ = \frac{4 πn}{3}

解

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

\frac{3 θ}{2} = π + 2 πn

在两边乘以

2

\frac{3 θ}{2} = π + 2 πn

在两边乘以

2

\frac{2 \cdot 3 θ}{2} = 2 π + 2 \cdot 2 πn

化简

3 θ = 2 π + 4 πn

3 θ = 2 π + 4 πn

两边除以

3

3 θ = 2 π + 4 πn

两边除以

3

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

化简

θ = \frac{2 π}{3} + \frac{4 πn}{3}

θ = \frac{2 π}{3} + \frac{4 πn}{3}

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

合并所有解

θ = \frac{π}{9} + \frac{4 πn}{9}, θ = \frac{π}{3} + \frac{4 πn}{9}, θ = \frac{4 πn}{3}, θ = \frac{2 π}{3} + \frac{4 πn}{3}

作图

流行的例子

sec(x)=-2/(sqrt(3))

sec (x) = - \frac{2}{3}

csc (2 x) = - 2

tan (θ) = \frac{13}{9}

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

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

sin(2x)=cos(x),0<= x<= 2pi

sin (2 x) = cos (x), 0 \leq x \leq 2 π