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受欢迎的三角函数 >

cos(5x)+cos(x)=2cos(2x)

解答

cos (5 x) + cos (x) = 2 cos (2 x)

解答

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

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n, x = 0^{\circ} + 12 0^{\circ} n

求解步骤

cos (5 x) + cos (x) = 2 cos (2 x)

两边减去

2 cos (2 x)

cos (5 x) + cos (x) - 2 cos (2 x) = 0

使用三角恒等式改写

cos (5 x) + cos (x) - 2 cos (2 x)

使用和差化积恒等式:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

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

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

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

\frac{5 x + x}{2} = 3 x

\frac{5 x + x}{2}

同类项相加:

5 x + x = 6 x

= \frac{6 x}{2}

数字相除:

\frac{6}{2} = 3

= 3 x

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

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

\frac{5 x - x}{2}

同类项相加:

5 x - x = 4 x

= \frac{4 x}{2}

数字相除:

\frac{4}{2} = 2

= 2 x

= 2 cos (3 x) cos (2 x)

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

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

分解

- 2 cos (2 x) + 2 cos (2 x) cos (3 x) : 2 cos (2 x) (cos (3 x) - 1)

- 2 cos (2 x) + 2 cos (2 x) cos (3 x)

因式分解出通项

2 cos (2 x)

= 2 cos (2 x) (- 1 + cos (3 x))

2 cos (2 x) (cos (3 x) - 1) = 0

分别求解每个部分

cos (2 x) = 0 or cos (3 x) - 1 = 0

cos (2 x) = 0 : x = \frac{π}{4} + πn, x = \frac{3 π}{4} + πn

cos (2 x) = 0

cos (2 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}

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

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

解

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

2 x = \frac{π}{2} + 2 πn

两边除以

2

2 x = \frac{π}{2} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

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

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

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

使用分式法则:

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

= \frac{π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

解

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

2 x = \frac{3 π}{2} + 2 πn

两边除以

2

2 x = \frac{3 π}{2} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

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

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

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

使用分式法则:

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

= \frac{3 π}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{3 π}{4}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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

cos (3 x) - 1 = 0 : x = \frac{2 πn}{3}

cos (3 x) - 1 = 0

将

1

到右边

cos (3 x) - 1 = 0

两边加上

1

cos (3 x) - 1 + 1 = 0 + 1

化简

cos (3 x) = 1

cos (3 x) = 1

cos (3 x) = 1

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}

3 x = 0 + 2 πn

3 x = 0 + 2 πn

解

3 x = 0 + 2 πn : x = \frac{2 πn}{3}

3 x = 0 + 2 πn

0 + 2 πn = 2 πn

3 x = 2 πn

两边除以

3

3 x = 2 πn

两边除以

3

\frac{3 x}{3} = \frac{2 πn}{3}

化简

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

x = \frac{2 πn}{3}

合并所有解

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

作图

流行的例子

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sin(x)=(1.2)/(1.6)

sin (x) = \frac{1.2}{1.6}

sin(x)=(14.9)/(19.9725)

sin (x) = \frac{14.9}{19.9725}

sin(θ)= 1/2 ,0<= θ<= 4pi

sin (θ) = \frac{1}{2}, 0 \leq θ \leq 4 π

3 - cosh (x) = 0