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受欢迎的三角函数 >

6cos(2x)-cos(x)-1=0

解答

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

解答

x = 0.63247 \dots + 2 πn, x = 2 π - 0.63247 \dots + 2 πn, x = 2.37926 \dots + 2 πn, x = - 2.37926 \dots + 2 πn

+1

度数

x = 36.23833 \dots^{\circ} + 36 0^{\circ} n, x = 323.76166 \dots^{\circ} + 36 0^{\circ} n, x = 136.32194 \dots^{\circ} + 36 0^{\circ} n, x = - 136.32194 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- 1 - cos (x) + 6 cos (2 x)

使用倍角公式:

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

= - 1 - cos (x) + 6 (2 cos^{2} (x) - 1)

化简

- 1 - cos (x) + 6 (2 cos^{2} (x) - 1) : 12 cos^{2} (x) - cos (x) - 7

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

乘开

6 (2 cos^{2} (x) - 1) : 12 cos^{2} (x) - 6

6 (2 cos^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = 6, b = 2 cos^{2} (x), c = 1

= 6 \cdot 2 cos^{2} (x) - 6 \cdot 1

化简

6 \cdot 2 cos^{2} (x) - 6 \cdot 1 : 12 cos^{2} (x) - 6

6 \cdot 2 cos^{2} (x) - 6 \cdot 1

数字相乘:

6 \cdot 2 = 12

= 12 cos^{2} (x) - 6 \cdot 1

数字相乘:

6 \cdot 1 = 6

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

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

= - 1 - cos (x) + 12 cos^{2} (x) - 6

化简

- 1 - cos (x) + 12 cos^{2} (x) - 6 : 12 cos^{2} (x) - cos (x) - 7

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

对同类项分组

= - cos (x) + 12 cos^{2} (x) - 1 - 6

数字相减:

- 1 - 6 = - 7

= 12 cos^{2} (x) - cos (x) - 7

= 12 cos^{2} (x) - cos (x) - 7

= 12 cos^{2} (x) - cos (x) - 7

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

用替代法求解

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

令

: cos (x) = u

- 7 - u + 12 u^{2} = 0

- 7 - u + 12 u^{2} = 0 : u = \frac{1 + 337}{24}, u = \frac{1 - 337}{24}

- 7 - u + 12 u^{2} = 0

改写成标准形式

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

12 u^{2} - u - 7 = 0

使用求根公式求解

12 u^{2} - u - 7 = 0

二次方程求根公式:

若

a = 12, b = - 1, c = - 7

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 12 ( - 7 )}{2 \cdot 12}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 12 ( - 7 )}{2 \cdot 12}

(- 1)^{2} - 4 \cdot 12 (- 7) = 337

(- 1)^{2} - 4 \cdot 12 (- 7)

使用法则

- (- a) = a

= (- 1)^{2} + 4 \cdot 12 \cdot 7

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 12 \cdot 7 = 336

4 \cdot 12 \cdot 7

数字相乘:

4 \cdot 12 \cdot 7 = 336

= 336

= 1 + 336

数字相加:

1 + 336 = 337

= 337

u_{1, 2} = \frac{- ( - 1 ) \pm 337}{2 \cdot 12}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 337}{2 \cdot 12}, u_{2} = \frac{- ( - 1 ) - 337}{2 \cdot 12}

u = \frac{- ( - 1 ) + 337}{2 \cdot 12} : \frac{1 + 337}{24}

\frac{- ( - 1 ) + 337}{2 \cdot 12}

使用法则

- (- a) = a

= \frac{1 + 337}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{1 + 337}{24}

u = \frac{- ( - 1 ) - 337}{2 \cdot 12} : \frac{1 - 337}{24}

\frac{- ( - 1 ) - 337}{2 \cdot 12}

使用法则

- (- a) = a

= \frac{1 - 337}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{1 - 337}{24}

二次方程组的解是:

u = \frac{1 + 337}{24}, u = \frac{1 - 337}{24}

u = cos (x)

代回

cos (x) = \frac{1 + 337}{24}, cos (x) = \frac{1 - 337}{24}

cos (x) = \frac{1 + 337}{24}, cos (x) = \frac{1 - 337}{24}

cos (x) = \frac{1 + 337}{24} : x = arccos (\frac{1 + 337}{24}) + 2 πn, x = 2 π - arccos (\frac{1 + 337}{24}) + 2 πn

cos (x) = \frac{1 + 337}{24}

使用反三角函数性质

cos (x) = \frac{1 + 337}{24}

cos (x) = \frac{1 + 337}{24}

的通解

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{1 + 337}{24}) + 2 πn, x = 2 π - arccos (\frac{1 + 337}{24}) + 2 πn

x = arccos (\frac{1 + 337}{24}) + 2 πn, x = 2 π - arccos (\frac{1 + 337}{24}) + 2 πn

cos (x) = \frac{1 - 337}{24} : x = arccos (\frac{1 - 337}{24}) + 2 πn, x = - arccos (\frac{1 - 337}{24}) + 2 πn

cos (x) = \frac{1 - 337}{24}

使用反三角函数性质

cos (x) = \frac{1 - 337}{24}

cos (x) = \frac{1 - 337}{24}

的通解

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (\frac{1 - 337}{24}) + 2 πn, x = - arccos (\frac{1 - 337}{24}) + 2 πn

x = arccos (\frac{1 - 337}{24}) + 2 πn, x = - arccos (\frac{1 - 337}{24}) + 2 πn

合并所有解

x = arccos (\frac{1 + 337}{24}) + 2 πn, x = 2 π - arccos (\frac{1 + 337}{24}) + 2 πn, x = arccos (\frac{1 - 337}{24}) + 2 πn, x = - arccos (\frac{1 - 337}{24}) + 2 πn

以小数形式表示解

x = 0.63247 \dots + 2 πn, x = 2 π - 0.63247 \dots + 2 πn, x = 2.37926 \dots + 2 πn, x = - 2.37926 \dots + 2 πn

作图

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