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受欢迎的三角函数 >

cos(3x)=cos(6x)

解答

cos (3 x) = cos (6 x)

解答

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

+1

度数

x = 0^{\circ} + 24 0^{\circ} n, x = 12 0^{\circ} + 24 0^{\circ} n, x = 0^{\circ} + 8 0^{\circ} n, x = 4 0^{\circ} + 8 0^{\circ} n

求解步骤

cos (3 x) = cos (6 x)

两边减去

cos (6 x)

cos (3 x) - cos (6 x) = 0

使用三角恒等式改写

cos (3 x) - cos (6 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})

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

化简

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

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

同类项相加:

3 x + 6 x = 9 x

= - 2 sin (\frac{9 x}{2}) sin (\frac{3 x - 6 x}{2})

\frac{3 x - 6 x}{2} = - \frac{3 x}{2}

\frac{3 x - 6 x}{2}

同类项相加:

3 x - 6 x = - 3 x

= \frac{- 3 x}{2}

使用分式法则:

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

= - \frac{3 x}{2}

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

使用负角恒等式:

sin (- x) = - sin (x)

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

使用法则

- (- a) = a

= 2 sin (\frac{3 x}{2}) sin (\frac{9 x}{2})

= 2 sin (\frac{3 x}{2}) sin (\frac{9 x}{2})

2 sin (\frac{3 x}{2}) sin (\frac{9 x}{2}) = 0

分别求解每个部分

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

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

sin (\frac{3 x}{2}) = 0

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

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

解

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

\frac{3 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 x}{2} = 2 πn

在两边乘以

2

\frac{3 x}{2} = 2 πn

在两边乘以

2

\frac{2 \cdot 3 x}{2} = 2 \cdot 2 πn

化简

3 x = 4 πn

3 x = 4 πn

两边除以

3

3 x = 4 πn

两边除以

3

\frac{3 x}{3} = \frac{4 πn}{3}

化简

x = \frac{4 πn}{3}

x = \frac{4 πn}{3}

解

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

\frac{3 x}{2} = π + 2 πn

在两边乘以

2

\frac{3 x}{2} = π + 2 πn

在两边乘以

2

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

化简

3 x = 2 π + 4 πn

3 x = 2 π + 4 πn

两边除以

3

3 x = 2 π + 4 πn

两边除以

3

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

化简

x = \frac{2 π}{3} + \frac{4 πn}{3}

x = \frac{2 π}{3} + \frac{4 πn}{3}

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

sin (\frac{9 x}{2}) = 0 : x = \frac{4 πn}{9}, x = \frac{2 π}{9} + \frac{4 πn}{9}

sin (\frac{9 x}{2}) = 0

sin (\frac{9 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{9 x}{2} = 0 + 2 πn, \frac{9 x}{2} = π + 2 πn

\frac{9 x}{2} = 0 + 2 πn, \frac{9 x}{2} = π + 2 πn

解

\frac{9 x}{2} = 0 + 2 πn : x = \frac{4 πn}{9}

\frac{9 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{9 x}{2} = 2 πn

在两边乘以

2

\frac{9 x}{2} = 2 πn

在两边乘以

2

\frac{2 \cdot 9 x}{2} = 2 \cdot 2 πn

化简

9 x = 4 πn

9 x = 4 πn

两边除以

9

9 x = 4 πn

两边除以

9

\frac{9 x}{9} = \frac{4 πn}{9}

化简

x = \frac{4 πn}{9}

x = \frac{4 πn}{9}

解

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

\frac{9 x}{2} = π + 2 πn

在两边乘以

2

\frac{9 x}{2} = π + 2 πn

在两边乘以

2

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

化简

9 x = 2 π + 4 πn

9 x = 2 π + 4 πn

两边除以

9

9 x = 2 π + 4 πn

两边除以

9

\frac{9 x}{9} = \frac{2 π}{9} + \frac{4 πn}{9}

化简

x = \frac{2 π}{9} + \frac{4 πn}{9}

x = \frac{2 π}{9} + \frac{4 πn}{9}

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

合并所有解

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

作图

流行的例子

cos (A) = \frac{5}{9}

cos (x) = 0.56

3 - cot^{2} (x) = 0

(-4cos(x)+2sin(x))^2+12sin^2(x)=24

(- 4 cos (x) + 2 sin (x))^{2} + 12 sin^{2} (x) = 24

-arccos(x)+arctan(5/12)=0

- arccos (x) + arctan (\frac{5}{12}) = 0