zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

cos(x)-cos(7x)=sin(4x)

解答

cos (x) - cos (7 x) = sin (4 x)

解答

x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}, x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

+1

度数

x = 0^{\circ} + 9 0^{\circ} n, x = 4 5^{\circ} + 9 0^{\circ} n, x = 1 0^{\circ} + 12 0^{\circ} n, x = 5 0^{\circ} + 12 0^{\circ} n

求解步骤

cos (x) - cos (7 x) = sin (4 x)

两边减去

sin (4 x)

cos (x) - cos (7 x) - sin (4 x) = 0

使用三角恒等式改写

- cos (7 x) + cos (x) - sin (4 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})

= - sin (4 x) - 2 sin (\frac{x + 7 x}{2}) sin (\frac{x - 7 x}{2})

化简

- sin (4 x) - 2 sin (\frac{x + 7 x}{2}) sin (\frac{x - 7 x}{2}) : - sin (4 x) + 2 sin (3 x) sin (4 x)

- sin (4 x) - 2 sin (\frac{x + 7 x}{2}) sin (\frac{x - 7 x}{2})

2 sin (\frac{x + 7 x}{2}) sin (\frac{x - 7 x}{2}) = - 2 sin (3 x) sin (4 x)

2 sin (\frac{x + 7 x}{2}) sin (\frac{x - 7 x}{2})

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

\frac{x + 7 x}{2}

同类项相加:

x + 7 x = 8 x

= \frac{8 x}{2}

数字相除:

\frac{8}{2} = 4

= 4 x

= 2 sin (4 x) sin (\frac{x - 7 x}{2})

\frac{x - 7 x}{2} = - 3 x

\frac{x - 7 x}{2}

同类项相加:

x - 7 x = - 6 x

= \frac{- 6 x}{2}

使用分式法则:

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

= - \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= - 3 x

= 2 sin (4 x) sin (- 3 x)

使用负角恒等式:

sin (- x) = - sin (x)

= 2 (- sin (3 x)) sin (4 x)

= - sin (4 x) - (- 2 sin (3 x) sin (4 x))

使用法则

- (- a) = a

= - sin (4 x) + 2 sin (3 x) sin (4 x)

= - sin (4 x) + 2 sin (3 x) sin (4 x)

- sin (4 x) + 2 sin (3 x) sin (4 x) = 0

分解

- sin (4 x) + 2 sin (3 x) sin (4 x) : sin (4 x) (2 sin (3 x) - 1)

- sin (4 x) + 2 sin (3 x) sin (4 x)

因式分解出通项

sin (4 x)

= sin (4 x) (- 1 + 2 sin (3 x))

sin (4 x) (2 sin (3 x) - 1) = 0

分别求解每个部分

sin (4 x) = 0 or 2 sin (3 x) - 1 = 0

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

sin (4 x) = 0

sin (4 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}

4 x = 0 + 2 πn, 4 x = π + 2 πn

4 x = 0 + 2 πn, 4 x = π + 2 πn

解

4 x = 0 + 2 πn : x = \frac{πn}{2}

4 x = 0 + 2 πn

0 + 2 πn = 2 πn

4 x = 2 πn

两边除以

4

4 x = 2 πn

两边除以

4

\frac{4 x}{4} = \frac{2 πn}{4}

化简

x = \frac{πn}{2}

x = \frac{πn}{2}

解

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

4 x = π + 2 πn

两边除以

4

4 x = π + 2 πn

两边除以

4

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

化简

x = \frac{π}{4} + \frac{πn}{2}

x = \frac{π}{4} + \frac{πn}{2}

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

2 sin (3 x) - 1 = 0 : x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

2 sin (3 x) - 1 = 0

将

1

到右边

2 sin (3 x) - 1 = 0

两边加上

1

2 sin (3 x) - 1 + 1 = 0 + 1

化简

2 sin (3 x) = 1

2 sin (3 x) = 1

两边除以

2

2 sin (3 x) = 1

两边除以

2

\frac{2 sin ( 3 x )}{2} = \frac{1}{2}

化简

sin (3 x) = \frac{1}{2}

sin (3 x) = \frac{1}{2}

sin (3 x) = \frac{1}{2}

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}

3 x = \frac{π}{6} + 2 πn, 3 x = \frac{5 π}{6} + 2 πn

3 x = \frac{π}{6} + 2 πn, 3 x = \frac{5 π}{6} + 2 πn

解

3 x = \frac{π}{6} + 2 πn : x = \frac{π}{18} + \frac{2 πn}{3}

3 x = \frac{π}{6} + 2 πn

两边除以

3

3 x = \frac{π}{6} + 2 πn

两边除以

3

\frac{3 x}{3} = \frac{\frac{π}{6}}{3} + \frac{2 πn}{3}

化简

\frac{3 x}{3} = \frac{\frac{π}{6}}{3} + \frac{2 πn}{3}

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{\frac{π}{6}}{3} + \frac{2 πn}{3} : \frac{π}{18} + \frac{2 πn}{3}

\frac{\frac{π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{6}}{3} = \frac{π}{18}

\frac{\frac{π}{6}}{3}

使用分式法则:

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

= \frac{π}{6 \cdot 3}

数字相乘:

6 \cdot 3 = 18

= \frac{π}{18}

= \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

解

3 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{18} + \frac{2 πn}{3}

3 x = \frac{5 π}{6} + 2 πn

两边除以

3

3 x = \frac{5 π}{6} + 2 πn

两边除以

3

\frac{3 x}{3} = \frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

化简

\frac{3 x}{3} = \frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3} : \frac{5 π}{18} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} = \frac{5 π}{18}

\frac{\frac{5 π}{6}}{3}

使用分式法则:

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

= \frac{5 π}{6 \cdot 3}

数字相乘:

6 \cdot 3 = 18

= \frac{5 π}{18}

= \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

合并所有解

x = \frac{πn}{2}, x = \frac{π}{4} + \frac{πn}{2}, x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

作图

流行的例子

cos (\frac{θ}{2}) = 1

csc^2(θ)+csc(θ)-2=0

csc^{2} (θ) + csc (θ) - 2 = 0

2 sin (x) = cos (x)

2+2sin(θ)=4cos^2(θ)

2 + 2 sin (θ) = 4 cos^{2} (θ)

9tan(x)sin(x)-5sin(x)=0

9 tan (x) sin (x) - 5 sin (x) = 0