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受欢迎的三角函数 >

cos(4x)-4cos(2x)=5

解答

cos (4 x) - 4 cos (2 x) = 5

解答

x = \frac{π}{2} + πn

+1

度数

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

求解步骤

cos (4 x) - 4 cos (2 x) = 5

两边减去

5

cos (4 x) - 4 cos (2 x) - 5 = 0

使用三角恒等式改写

- 5 + cos (4 x) - 4 cos (2 x)

cos (4 x) = 2 cos^{2} (2 x) - 1

cos (4 x)

改写为

= cos (2 \cdot 2 x)

使用倍角公式:

cos (2 x) = 2 cos^{2} (x) - 1

cos (2 \cdot 2 x) = 2 cos^{2} (2 x) - 1

= 2 cos^{2} (2 x) - 1

= - 5 + 2 cos^{2} (2 x) - 1 - 4 cos (2 x)

化简

- 5 + 2 cos^{2} (2 x) - 1 - 4 cos (2 x) : 2 cos^{2} (2 x) - 4 cos (2 x) - 6

- 5 + 2 cos^{2} (2 x) - 1 - 4 cos (2 x)

对同类项分组

= 2 cos^{2} (2 x) - 4 cos (2 x) - 5 - 1

数字相减:

- 5 - 1 = - 6

= 2 cos^{2} (2 x) - 4 cos (2 x) - 6

= 2 cos^{2} (2 x) - 4 cos (2 x) - 6

- 6 + 2 cos^{2} (2 x) - 4 cos (2 x) = 0

用替代法求解

- 6 + 2 cos^{2} (2 x) - 4 cos (2 x) = 0

令

: cos (2 x) = u

- 6 + 2 u^{2} - 4 u = 0

- 6 + 2 u^{2} - 4 u = 0 : u = 3, u = - 1

- 6 + 2 u^{2} - 4 u = 0

改写成标准形式

a x^{2} + b x + c = 0

2 u^{2} - 4 u - 6 = 0

使用求根公式求解

2 u^{2} - 4 u - 6 = 0

二次方程求根公式:

若

a = 2, b = - 4, c = - 6

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 2 ( - 6 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 2 ( - 6 )}{2 \cdot 2}

(- 4)^{2} - 4 \cdot 2 (- 6) = 8

(- 4)^{2} - 4 \cdot 2 (- 6)

使用法则

- (- a) = a

= (- 4)^{2} + 4 \cdot 2 \cdot 6

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 4)^{2} = 4^{2}

= 4^{2} + 4 \cdot 2 \cdot 6

数字相乘:

4 \cdot 2 \cdot 6 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数字相加:

16 + 48 = 64

= 64

因式分解数字:

64 = 8^{2}

= 8^{2}

使用根式运算法则:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 4 ) \pm 8}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 4 ) + 8}{2 \cdot 2}, u_{2} = \frac{- ( - 4 ) - 8}{2 \cdot 2}

u = \frac{- ( - 4 ) + 8}{2 \cdot 2} : 3

\frac{- ( - 4 ) + 8}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{4 + 8}{2 \cdot 2}

数字相加:

4 + 8 = 12

= \frac{12}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 4 ) - 8}{2 \cdot 2} : - 1

\frac{- ( - 4 ) - 8}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{4 - 8}{2 \cdot 2}

数字相减:

4 - 8 = - 4

= \frac{- 4}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{4}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{4}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = 3, u = - 1

u = cos (2 x)

代回

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

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

cos (2 x) = 3 :

无解

cos (2 x) = 3

- 1 \leq cos (x) \leq 1

无解

cos (2 x) = - 1 : x = \frac{π}{2} + πn

cos (2 x) = - 1

cos (2 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}

2 x = π + 2 πn

2 x = π + 2 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 = \frac{π}{2} + πn

合并所有解

x = \frac{π}{2} + πn

作图

流行的例子

sin(x)=sin(x+(-3)pi)

sin (x) = sin (x + (- 3) π)

sin (2 x) - \frac{2}{3} = 0

2sin(θ)+cos(2θ)=0

2 sin (θ) + cos (2 θ) = 0

sin(x+pi/4)=sqrt(2)cos(x+pi/4)

sin (x + \frac{π}{4}) = 2 cos (x + \frac{π}{4})

5 = 7 cos (\frac{π}{3} t)