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受欢迎的三角函数 >

2cos(2x)=4cos(x)

解答

2 cos (2 x) = 4 cos (x)

解答

x = 1.94553 \dots + 2 πn, x = - 1.94553 \dots + 2 πn

+1

度数

x = 111.47070 \dots^{\circ} + 36 0^{\circ} n, x = - 111.47070 \dots^{\circ} + 36 0^{\circ} n

求解步骤

2 cos (2 x) = 4 cos (x)

两边减去

4 cos (x)

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

使用三角恒等式改写

2 cos (2 x) - 4 cos (x)

使用倍角公式:

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

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

(- 1 + 2 cos^{2} (x)) \cdot 2 - 4 cos (x) = 0

用替代法求解

(- 1 + 2 cos^{2} (x)) \cdot 2 - 4 cos (x) = 0

令

: cos (x) = u

(- 1 + 2 u^{2}) \cdot 2 - 4 u = 0

(- 1 + 2 u^{2}) \cdot 2 - 4 u = 0 : u = \frac{1 + 3}{2}, u = \frac{1 - 3}{2}

(- 1 + 2 u^{2}) \cdot 2 - 4 u = 0

展开

(- 1 + 2 u^{2}) \cdot 2 - 4 u : - 2 + 4 u^{2} - 4 u

(- 1 + 2 u^{2}) \cdot 2 - 4 u

= 2 (- 1 + 2 u^{2}) - 4 u

乘开

2 (- 1 + 2 u^{2}) : - 2 + 4 u^{2}

2 (- 1 + 2 u^{2})

使用分配律:

a (b + c) = ab + a c

a = 2, b = - 1, c = 2 u^{2}

= 2 (- 1) + 2 \cdot 2 u^{2}

使用加减运算法则

+ (- a) = - a

= - 2 \cdot 1 + 2 \cdot 2 u^{2}

化简

- 2 \cdot 1 + 2 \cdot 2 u^{2} : - 2 + 4 u^{2}

- 2 \cdot 1 + 2 \cdot 2 u^{2}

数字相乘:

2 \cdot 1 = 2

= - 2 + 2 \cdot 2 u^{2}

数字相乘:

2 \cdot 2 = 4

= - 2 + 4 u^{2}

= - 2 + 4 u^{2}

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

(- 4)^{2} - 4 \cdot 4 (- 2) = 43

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 4 \cdot 2 = 32

= 4^{2} + 32

4^{2} = 16

= 16 + 32

数字相加:

16 + 32 = 48

= 48

48

质因数分解:

2^{4} \cdot 3

48

48

除以

248 = 24 \cdot 2

= 2 \cdot 24

24

除以

224 = 12 \cdot 2

= 2 \cdot 2 \cdot 12

12

除以

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

= 2^{4} \cdot 3

= 2^{4} \cdot 3

使用根式运算法则:

n ab = n a n b

= 3 2^{4}

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}}

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 3

整理后得

= 43

u_{1, 2} = \frac{- ( - 4 ) \pm 4 3}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 4 ) + 4 3}{2 \cdot 4}, u_{2} = \frac{- ( - 4 ) - 4 3}{2 \cdot 4}

u = \frac{- ( - 4 ) + 4 3}{2 \cdot 4} : \frac{1 + 3}{2}

\frac{- ( - 4 ) + 4 3}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{4 + 4 3}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{4 + 4 3}{8}

分解

4 + 43 : 4 (1 + 3)

4 + 43

改写为

= 4 \cdot 1 + 43

因式分解出通项

4

= 4 (1 + 3)

= \frac{4 ( 1 + 3 )}{8}

约分:

4

= \frac{1 + 3}{2}

u = \frac{- ( - 4 ) - 4 3}{2 \cdot 4} : \frac{1 - 3}{2}

\frac{- ( - 4 ) - 4 3}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{4 - 4 3}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{4 - 4 3}{8}

分解

4 - 43 : 4 (1 - 3)

4 - 43

改写为

= 4 \cdot 1 - 43

因式分解出通项

4

= 4 (1 - 3)

= \frac{4 ( 1 - 3 )}{8}

约分:

4

= \frac{1 - 3}{2}

二次方程组的解是:

u = \frac{1 + 3}{2}, u = \frac{1 - 3}{2}

u = cos (x)

代回

cos (x) = \frac{1 + 3}{2}, cos (x) = \frac{1 - 3}{2}

cos (x) = \frac{1 + 3}{2}, cos (x) = \frac{1 - 3}{2}

cos (x) = \frac{1 + 3}{2} :

无解

cos (x) = \frac{1 + 3}{2}

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{1 - 3}{2} : x = arccos (\frac{1 - 3}{2}) + 2 πn, x = - arccos (\frac{1 - 3}{2}) + 2 πn

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

使用反三角函数性质

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

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

的通解

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

x = arccos (\frac{1 - 3}{2}) + 2 πn, x = - arccos (\frac{1 - 3}{2}) + 2 πn

x = arccos (\frac{1 - 3}{2}) + 2 πn, x = - arccos (\frac{1 - 3}{2}) + 2 πn

合并所有解

x = arccos (\frac{1 - 3}{2}) + 2 πn, x = - arccos (\frac{1 - 3}{2}) + 2 πn

以小数形式表示解

x = 1.94553 \dots + 2 πn, x = - 1.94553 \dots + 2 πn

作图

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