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受欢迎的三角函数 >

cos(3x)-cos(2x)=0

解答

cos (3 x) - cos (2 x) = 0

解答

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

+1

度数

x = 0^{\circ} + 14 4^{\circ} n, x = 7 2^{\circ} + 14 4^{\circ} n

求解步骤

cos (3 x) - cos (2 x) = 0

使用三角恒等式改写

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

化简

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

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

同类项相加:

3 x + 2 x = 5 x

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

同类项相加:

3 x - 2 x = x

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

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

- 2 sin (\frac{5 x}{2}) sin (\frac{x}{2}) = 0

分别求解每个部分

sin (\frac{5 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{5 x}{2}) = 0 : x = \frac{4 πn}{5}, x = \frac{2 π}{5} + \frac{4 πn}{5}

sin (\frac{5 x}{2}) = 0

sin (\frac{5 x}{2}) = 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}

\frac{5 x}{2} = 0 + 2 πn, \frac{5 x}{2} = π + 2 πn

\frac{5 x}{2} = 0 + 2 πn, \frac{5 x}{2} = π + 2 πn

解

\frac{5 x}{2} = 0 + 2 πn : x = \frac{4 πn}{5}

\frac{5 x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{5 x}{2} = 2 πn

在两边乘以

2

\frac{5 x}{2} = 2 πn

在两边乘以

2

\frac{2 \cdot 5 x}{2} = 2 \cdot 2 πn

化简

5 x = 4 πn

5 x = 4 πn

两边除以

5

5 x = 4 πn

两边除以

5

\frac{5 x}{5} = \frac{4 πn}{5}

化简

x = \frac{4 πn}{5}

x = \frac{4 πn}{5}

解

\frac{5 x}{2} = π + 2 πn : x = \frac{2 π}{5} + \frac{4 πn}{5}

\frac{5 x}{2} = π + 2 πn

在两边乘以

2

\frac{5 x}{2} = π + 2 πn

在两边乘以

2

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

化简

5 x = 2 π + 4 πn

5 x = 2 π + 4 πn

两边除以

5

5 x = 2 π + 4 πn

两边除以

5

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

化简

x = \frac{2 π}{5} + \frac{4 πn}{5}

x = \frac{2 π}{5} + \frac{4 πn}{5}

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

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

sin (\frac{x}{2}) = 0

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

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

解

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

在两边乘以

2

\frac{x}{2} = 2 πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot 2 πn

化简

x = 4 πn

x = 4 πn

解

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

\frac{x}{2} = π + 2 πn

在两边乘以

2

\frac{x}{2} = π + 2 πn

在两边乘以

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

化简

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

合并所有解

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

合并重叠的区间

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

作图

流行的例子

tan (2 x + 1 0^{\circ}) = 1.5

tan(x+20)=cot(x)

tan (x + 2 0^{\circ}) = cot (x)

3 csc^{2} (x) + 1 = 5

6cos^2(x)-12cos(x)+6=0

6 cos^{2} (x) - 12 cos (x) + 6 = 0

6sin^2(θ)-10sin(θ)-8=-3sin(θ)-3

6 sin^{2} (θ) - 10 sin (θ) - 8 = - 3 sin (θ) - 3