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受欢迎的三角函数 >

cos(3x)-cos(x)=0

解答

cos (3 x) - cos (x) = 0

解答

x = πn, x = \frac{π}{2} + πn

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n

求解步骤

cos (3 x) - cos (x) = 0

使用三角恒等式改写

cos (3 x) - cos (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})

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

化简

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

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

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

\frac{3 x + x}{2}

同类项相加:

3 x + x = 4 x

= \frac{4 x}{2}

数字相除:

\frac{4}{2} = 2

= 2 x

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

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

\frac{3 x - x}{2}

同类项相加:

3 x - x = 2 x

= \frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

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

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

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

分别求解每个部分

sin (2 x) = 0 or sin (x) = 0

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

sin (2 x) = 0

sin (2 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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

解

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

两边除以

2

2 x = 2 πn

两边除以

2

\frac{2 x}{2} = \frac{2 πn}{2}

化简

x = πn

x = πn

解

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

两边除以

2

2 x = π + 2 πn

两边除以

2

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

化简

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

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

合并重叠的区间

x = πn, x = \frac{π}{2} + πn

作图

流行的例子

7sin(x)cos(x)-10cos(x)=0

7 sin (x) cos (x) - 10 cos (x) = 0

sin(pi-3x)= 1/2

sin (π - 3 x) = \frac{1}{2}

sin (x) = 0.25

tan(x)(tan(x)+1)=0

tan (x) (tan (x) + 1) = 0

cos(θ)=sin(2θ)

cos (θ) = sin (2 θ)