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受欢迎的三角函数 >

sin(5x)=sin(7x)

解答

sin (5 x) = sin (7 x)

解答

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}, x = 2 πn, x = π + 2 πn

+1

度数

x = 1 5^{\circ} + 6 0^{\circ} n, x = 4 5^{\circ} + 6 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

sin (5 x) = sin (7 x)

两边减去

sin (7 x)

sin (5 x) - sin (7 x) = 0

使用三角恒等式改写

sin (5 x) - sin (7 x)

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

化简

2 sin (\frac{5 x - 7 x}{2}) cos (\frac{5 x + 7 x}{2}) : - 2 cos (6 x) sin (x)

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

\frac{5 x - 7 x}{2} = - x

\frac{5 x - 7 x}{2}

同类项相加:

5 x - 7 x = - 2 x

= \frac{- 2 x}{2}

使用分式法则:

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

= - \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= - x

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

使用负角恒等式:

sin (- x) = - sin (x)

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

去除括号:

(- a) = - a

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

\frac{5 x + 7 x}{2} = 6 x

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

同类项相加:

5 x + 7 x = 12 x

= \frac{12 x}{2}

数字相除:

\frac{12}{2} = 6

= 6 x

= - 2 cos (6 x) sin (x)

= - 2 cos (6 x) sin (x)

- 2 cos (6 x) sin (x) = 0

分别求解每个部分

cos (6 x) = 0 or sin (x) = 0

cos (6 x) = 0 : x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

cos (6 x) = 0

cos (6 x) = 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}

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

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

解

6 x = \frac{π}{2} + 2 πn : x = \frac{π}{12} + \frac{πn}{3}

6 x = \frac{π}{2} + 2 πn

两边除以

6

6 x = \frac{π}{2} + 2 πn

两边除以

6

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

化简

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

化简

\frac{6 x}{6} : x

\frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{π}{2}}{6} + \frac{2 πn}{6} : \frac{π}{12} + \frac{πn}{3}

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

\frac{\frac{π}{2}}{6} = \frac{π}{12}

\frac{\frac{π}{2}}{6}

使用分式法则:

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

= \frac{π}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{π}{12}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

约分:

2

= \frac{πn}{3}

= \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}

解

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

6 x = \frac{3 π}{2} + 2 πn

两边除以

6

6 x = \frac{3 π}{2} + 2 πn

两边除以

6

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

化简

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

化简

\frac{6 x}{6} : x

\frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{3 π}{2}}{6} + \frac{2 πn}{6} : \frac{π}{4} + \frac{πn}{3}

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

\frac{\frac{3 π}{2}}{6} = \frac{π}{4}

\frac{\frac{3 π}{2}}{6}

使用分式法则:

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

= \frac{3 π}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{3 π}{12}

约分:

3

= \frac{π}{4}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

约分:

2

= \frac{πn}{3}

= \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

合并所有解

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}, x = 2 πn, x = π + 2 πn

作图

流行的例子

cos (x) = 0, 60

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

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

cos (x) = 0, 56

cos (\frac{x}{2}) = \frac{5}{6}

sin (B) = \frac{9}{16}