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受欢迎的三角函数 >

sin(4x)+sin(6x)=0

解答

sin (4 x) + sin (6 x) = 0

解答

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

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 7 2^{\circ} n, x = 3 6^{\circ} + 7 2^{\circ} n

求解步骤

sin (4 x) + sin (6 x) = 0

使用三角恒等式改写

sin (4 x) + sin (6 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{4 x + 6 x}{2}) cos (\frac{4 x - 6 x}{2})

化简

2 sin (\frac{4 x + 6 x}{2}) cos (\frac{4 x - 6 x}{2}) : 2 cos (x) sin (5 x)

2 sin (\frac{4 x + 6 x}{2}) cos (\frac{4 x - 6 x}{2})

\frac{4 x + 6 x}{2} = 5 x

\frac{4 x + 6 x}{2}

同类项相加:

4 x + 6 x = 10 x

= \frac{10 x}{2}

数字相除:

\frac{10}{2} = 5

= 5 x

= 2 sin (5 x) cos (\frac{4 x - 6 x}{2})

\frac{4 x - 6 x}{2} = - x

\frac{4 x - 6 x}{2}

同类项相加:

4 x - 6 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 (5 x) cos (- x)

使用负角恒等式:

cos (- x) = cos (x)

= 2 cos (x) sin (5 x)

= 2 cos (x) sin (5 x)

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

分别求解每个部分

cos (x) = 0 or sin (5 x) = 0

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

cos (x) = 0

cos (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}

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

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

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}

合并所有解

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

作图

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