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受欢迎的三角函数 >

cos(3x)-cos(7x)=0

解答

cos (3 x) - cos (7 x) = 0

解答

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

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 7 2^{\circ} n, x = 3 6^{\circ} + 7 2^{\circ} n

求解步骤

cos (3 x) - cos (7 x) = 0

使用三角恒等式改写

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

化简

- 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)

使用法则

- (- a) = a

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

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

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

分别求解每个部分

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

作图

流行的例子

2 sin (x - 1) = 0

tan (x) = 0.42

sin(3x)+sin(x)=sin(2x)

sin (3 x) + sin (x) = sin (2 x)

2cos^2(θ)-4=3cos(θ)-2

2 cos^{2} (θ) - 4 = 3 cos (θ) - 2

cos (x) = 1.2