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cos^2(x)+6cos(x)=-3,y=3

解答

cos^{2} (x) + 6 cos (x) = - 3, y = 3

解答

x \in R 无解

求解步骤

cos^{2} (x) + 6 cos (x) = - 3, y = 3

用替代法求解

cos^{2} (x) + 6 cos (x) = - 3

令

: cos (x) = u

u^{2} + 6 u = - 3

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

u^{2} + 6 u = - 3

将

3

para o lado esquerdo

u^{2} + 6 u = - 3

两边加上

3

u^{2} + 6 u + 3 = - 3 + 3

化简

u^{2} + 6 u + 3 = 0

u^{2} + 6 u + 3 = 0

使用求根公式求解

u^{2} + 6 u + 3 = 0

二次方程求根公式:

若

a = 1, b = 6, c = 3

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 1 \cdot 3}{2 \cdot 1}

6^{2} - 4 \cdot 1 \cdot 3 = 26

6^{2} - 4 \cdot 1 \cdot 3

数字相乘:

4 \cdot 1 \cdot 3 = 12

= 6^{2} - 12

6^{2} = 36

= 36 - 12

数字相减:

36 - 12 = 24

= 24

24

质因数分解:

2^{3} \cdot 3

24

24

除以

224 = 12 \cdot 2

= 2 \cdot 12

12

除以

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 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 3

= 2^{3} \cdot 3

= 2^{3} \cdot 3

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 2 \cdot 3

使用根式运算法则:

= 2^{2} 2 \cdot 3

使用根式运算法则:

2^{2} = 2

= 2 2 \cdot 3

整理后得

= 26

u_{1, 2} = \frac{- 6 \pm 2 6}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 6 + 2 6}{2 \cdot 1}, u_{2} = \frac{- 6 - 2 6}{2 \cdot 1}

u = \frac{- 6 + 2 6}{2 \cdot 1} : - 3 + 6

\frac{- 6 + 2 6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 + 2 6}{2}

分解

- 6 + 26 : 2 (- 3 + 6)

- 6 + 26

改写为

= - 2 \cdot 3 + 26

因式分解出通项

2

= 2 (- 3 + 6)

= \frac{2 ( - 3 + 6 )}{2}

数字相除:

\frac{2}{2} = 1

= - 3 + 6

u = \frac{- 6 - 2 6}{2 \cdot 1} : - 3 - 6

\frac{- 6 - 2 6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 - 2 6}{2}

分解

- 6 - 26 : - 2 (3 + 6)

- 6 - 26

改写为

= - 2 \cdot 3 - 26

因式分解出通项

2

= - 2 (3 + 6)

= - \frac{2 ( 3 + 6 )}{2}

数字相除:

\frac{2}{2} = 1

= - (3 + 6)

取负

- (3 + 6) = - 3 - 6

= - 3 - 6

二次方程组的解是:

u = - 3 + 6, u = - 3 - 6

u = cos (x)

代回

cos (x) = - 3 + 6, cos (x) = - 3 - 6

cos (x) = - 3 + 6, cos (x) = - 3 - 6

cos (x) = - 3 + 6, y = 3 :

无解

cos (x) = - 3 + 6, y = 3

使用反三角函数性质

cos (x) = - 3 + 6

cos (x) = - 3 + 6

的通解

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

x = arccos (- 3 + 6) + 2 πn, x = - arccos (- 3 + 6) + 2 πn

x = arccos (- 3 + 6) + 2 πn, x = - arccos (- 3 + 6) + 2 πn

在

y = 3

范围内的解

无解

cos (x) = - 3 - 6, y = 3 :

无解

cos (x) = - 3 - 6, y = 3

- 1 \leq cos (x) \leq 1

无解

合并所有解

x \in R 无解

流行的例子

2sec^2(x)-5tan(x)=5

2 sec^{2} (x) - 5 tan (x) = 5

coth (z) = 1

cos(10x)=sin(5x-3)

cos (10 x) = sin (5 x - 3)

13tan(a)=10,0b=100

13 tan (a) = 10, 0 b = 100

- 4 cos (3 x) = 0