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受欢迎的三角函数 >

cos(5x)=-cos(2x)

解答

cos (5 x) = - cos (2 x)

解答

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

+1

度数

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n, x = 25.71428 \dots^{\circ} + 102.85714 \dots^{\circ} n, x = 77.14285 \dots^{\circ} + 102.85714 \dots^{\circ} n

求解步骤

cos (5 x) = - cos (2 x)

两边减去

- cos (2 x)

cos (5 x) + cos (2 x) = 0

使用三角恒等式改写

cos (2 x) + cos (5 x)

使用和差化积恒等式:

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

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

化简

2 cos (\frac{2 x + 5 x}{2}) cos (\frac{2 x - 5 x}{2}) : 2 cos (\frac{3 x}{2}) cos (\frac{7 x}{2})

2 cos (\frac{2 x + 5 x}{2}) cos (\frac{2 x - 5 x}{2})

同类项相加:

2 x + 5 x = 7 x

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

\frac{2 x - 5 x}{2} = - \frac{3 x}{2}

\frac{2 x - 5 x}{2}

同类项相加:

2 x - 5 x = - 3 x

= \frac{- 3 x}{2}

使用分式法则:

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

= - \frac{3 x}{2}

= 2 cos (\frac{7 x}{2}) cos (- \frac{3 x}{2})

使用负角恒等式:

cos (- x) = cos (x)

= 2 cos (\frac{3 x}{2}) cos (\frac{7 x}{2})

= 2 cos (\frac{3 x}{2}) cos (\frac{7 x}{2})

2 cos (\frac{3 x}{2}) cos (\frac{7 x}{2}) = 0

分别求解每个部分

cos (\frac{3 x}{2}) = 0 or cos (\frac{7 x}{2}) = 0

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

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

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

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

解

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

\frac{3 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

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

化简

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

化简

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

数字相乘:

2 \cdot 3 = 6

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 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

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

两边除以

3

3 x = π + 4 πn

两边除以

3

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

化简

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

解

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

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

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

化简

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

化简

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

数字相乘:

2 \cdot 3 = 6

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 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

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

两边除以

3

3 x = 3 π + 4 πn

两边除以

3

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

化简

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

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

cos (\frac{7 x}{2}) = 0 : x = \frac{π}{7} + \frac{4 πn}{7}, x = \frac{3 π}{7} + \frac{4 πn}{7}

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

cos (\frac{7 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{7 x}{2} = \frac{π}{2} + 2 πn, \frac{7 x}{2} = \frac{3 π}{2} + 2 πn

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

解

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

\frac{7 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

\frac{7 x}{2} = \frac{π}{2} + 2 πn

在两边乘以

2

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

化简

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

化简

\frac{2 \cdot 7 x}{2} : 7 x

\frac{2 \cdot 7 x}{2}

数字相乘:

2 \cdot 7 = 14

= \frac{14 x}{2}

数字相除:

\frac{14}{2} = 7

= 7 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

7 x = π + 4 πn

7 x = π + 4 πn

7 x = π + 4 πn

两边除以

7

7 x = π + 4 πn

两边除以

7

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

化简

x = \frac{π}{7} + \frac{4 πn}{7}

x = \frac{π}{7} + \frac{4 πn}{7}

解

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

\frac{7 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{7 x}{2} = \frac{3 π}{2} + 2 πn

在两边乘以

2

\frac{2 \cdot 7 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 7 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

化简

\frac{2 \cdot 7 x}{2} : 7 x

\frac{2 \cdot 7 x}{2}

数字相乘:

2 \cdot 7 = 14

= \frac{14 x}{2}

数字相除:

\frac{14}{2} = 7

= 7 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

7 x = 3 π + 4 πn

7 x = 3 π + 4 πn

7 x = 3 π + 4 πn

两边除以

7

7 x = 3 π + 4 πn

两边除以

7

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

化简

x = \frac{3 π}{7} + \frac{4 πn}{7}

x = \frac{3 π}{7} + \frac{4 πn}{7}

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

合并所有解

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

作图

流行的例子

cos^2(x)-cos(x)-2=0

cos^{2} (x) - cos (x) - 2 = 0

5sin(x)=3sin(x)+sqrt(3)

5 sin (x) = 3 sin (x) + 3

9cot^2(θ)-16=0

9 cot^{2} (θ) - 16 = 0

cos (θ) = - \frac{3}{4}

sin^2(x)-3sin(x)+2=0

sin^{2} (x) - 3 sin (x) + 2 = 0