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受欢迎的三角函数 >

-cos(7x)+cos(3x)=sin(5x)

解答

- cos (7 x) + cos (3 x) = sin (5 x)

解答

x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}, x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

+1

度数

x = 0^{\circ} + 7 2^{\circ} n, x = 3 6^{\circ} + 7 2^{\circ} n, x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n

求解步骤

- cos (7 x) + cos (3 x) = sin (5 x)

两边减去

sin (5 x)

- cos (7 x) + cos (3 x) - sin (5 x) = 0

使用三角恒等式改写

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

化简

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

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

2 sin (\frac{3 x + 7 x}{2}) sin (\frac{3 x - 7 x}{2}) = - 2 sin (2 x) sin (5 x)

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

\frac{3 x + 7 x}{2} = 5 x

\frac{3 x + 7 x}{2}

同类项相加:

3 x + 7 x = 10 x

= \frac{10 x}{2}

数字相除:

\frac{10}{2} = 5

= 5 x

= 2 sin (5 x) sin (\frac{3 x - 7 x}{2})

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

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

同类项相加:

3 x - 7 x = - 4 x

= \frac{- 4 x}{2}

使用分式法则:

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

= - \frac{4 x}{2}

数字相除:

\frac{4}{2} = 2

= - 2 x

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

使用负角恒等式:

sin (- x) = - sin (x)

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

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

使用法则

- (- a) = a

= - sin (5 x) + 2 sin (2 x) sin (5 x)

= - sin (5 x) + 2 sin (2 x) sin (5 x)

- sin (5 x) + 2 sin (2 x) sin (5 x) = 0

分解

- sin (5 x) + 2 sin (2 x) sin (5 x) : sin (5 x) (2 sin (2 x) - 1)

- sin (5 x) + 2 sin (2 x) sin (5 x)

因式分解出通项

sin (5 x)

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

sin (5 x) (2 sin (2 x) - 1) = 0

分别求解每个部分

sin (5 x) = 0 or 2 sin (2 x) - 1 = 0

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

sin (5 x) = 0

sin (5 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}

5 x = 0 + 2 πn, 5 x = π + 2 πn

5 x = 0 + 2 πn, 5 x = π + 2 πn

解

5 x = 0 + 2 πn : x = \frac{2 πn}{5}

5 x = 0 + 2 πn

0 + 2 πn = 2 πn

5 x = 2 πn

两边除以

5

5 x = 2 πn

两边除以

5

\frac{5 x}{5} = \frac{2 πn}{5}

化简

x = \frac{2 πn}{5}

x = \frac{2 πn}{5}

解

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

5 x = π + 2 πn

两边除以

5

5 x = π + 2 πn

两边除以

5

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

化简

x = \frac{π}{5} + \frac{2 πn}{5}

x = \frac{π}{5} + \frac{2 πn}{5}

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

2 sin (2 x) - 1 = 0 : x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

2 sin (2 x) - 1 = 0

将

1

到右边

2 sin (2 x) - 1 = 0

两边加上

1

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

化简

2 sin (2 x) = 1

2 sin (2 x) = 1

两边除以

2

2 sin (2 x) = 1

两边除以

2

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

化简

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

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

sin (2 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}

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

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

解

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

两边除以

2

2 x = \frac{π}{6} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

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

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

使用分式法则:

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

= \frac{π}{6 \cdot 2}

数字相乘:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

解

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

两边除以

2

2 x = \frac{5 π}{6} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

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

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

使用分式法则:

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

= \frac{5 π}{6 \cdot 2}

数字相乘:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

合并所有解

x = \frac{2 πn}{5}, x = \frac{π}{5} + \frac{2 πn}{5}, x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

作图

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