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受欢迎的三角函数 >

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

解答

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

解答

x = 1.74886 \dots + 2 πn, x = - 1.74886 \dots + 2 πn

+1

度数

x = 100.20230 \dots^{\circ} + 36 0^{\circ} n, x = - 100.20230 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cos (x) = u

2 u^{2} + 6 u = - 1

2 u^{2} + 6 u = - 1 : u = \frac{- 3 + 7}{2}, u = - \frac{3 + 7}{2}

2 u^{2} + 6 u = - 1

将

1

para o lado esquerdo

2 u^{2} + 6 u = - 1

两边加上

1

2 u^{2} + 6 u + 1 = - 1 + 1

化简

2 u^{2} + 6 u + 1 = 0

2 u^{2} + 6 u + 1 = 0

使用求根公式求解

2 u^{2} + 6 u + 1 = 0

二次方程求根公式:

若

a = 2, b = 6, c = 1

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

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

6^{2} - 4 \cdot 2 \cdot 1 = 27

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

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 6^{2} - 8

6^{2} = 36

= 36 - 8

数字相减:

36 - 8 = 28

= 28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

u_{1, 2} = \frac{- 6 \pm 2 7}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 6 + 2 7}{2 \cdot 2}, u_{2} = \frac{- 6 - 2 7}{2 \cdot 2}

u = \frac{- 6 + 2 7}{2 \cdot 2} : \frac{- 3 + 7}{2}

\frac{- 6 + 2 7}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 6 + 2 7}{4}

分解

- 6 + 27 : 2 (- 3 + 7)

- 6 + 27

改写为

= - 2 \cdot 3 + 27

因式分解出通项

2

= 2 (- 3 + 7)

= \frac{2 ( - 3 + 7 )}{4}

约分:

2

= \frac{- 3 + 7}{2}

u = \frac{- 6 - 2 7}{2 \cdot 2} : - \frac{3 + 7}{2}

\frac{- 6 - 2 7}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 6 - 2 7}{4}

分解

- 6 - 27 : - 2 (3 + 7)

- 6 - 27

改写为

= - 2 \cdot 3 - 27

因式分解出通项

2

= - 2 (3 + 7)

= - \frac{2 ( 3 + 7 )}{4}

约分:

2

= - \frac{3 + 7}{2}

二次方程组的解是:

u = \frac{- 3 + 7}{2}, u = - \frac{3 + 7}{2}

u = cos (x)

代回

cos (x) = \frac{- 3 + 7}{2}, cos (x) = - \frac{3 + 7}{2}

cos (x) = \frac{- 3 + 7}{2}, cos (x) = - \frac{3 + 7}{2}

cos (x) = \frac{- 3 + 7}{2} : x = arccos (\frac{- 3 + 7}{2}) + 2 πn, x = - arccos (\frac{- 3 + 7}{2}) + 2 πn

cos (x) = \frac{- 3 + 7}{2}

使用反三角函数性质

cos (x) = \frac{- 3 + 7}{2}

cos (x) = \frac{- 3 + 7}{2}

的通解

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

x = arccos (\frac{- 3 + 7}{2}) + 2 πn, x = - arccos (\frac{- 3 + 7}{2}) + 2 πn

x = arccos (\frac{- 3 + 7}{2}) + 2 πn, x = - arccos (\frac{- 3 + 7}{2}) + 2 πn

cos (x) = - \frac{3 + 7}{2} :

无解

cos (x) = - \frac{3 + 7}{2}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{- 3 + 7}{2}) + 2 πn, x = - arccos (\frac{- 3 + 7}{2}) + 2 πn

以小数形式表示解

x = 1.74886 \dots + 2 πn, x = - 1.74886 \dots + 2 πn

作图

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