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受欢迎的三角函数 >

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

解答

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

解答

x = 1.10460 \dots + 2 πn, x = 2 π - 1.10460 \dots + 2 πn

+1

度数

x = 63.28904 \dots^{\circ} + 36 0^{\circ} n, x = 296.71095 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cos (x) = u

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

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = 4, c = - 2

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

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

4^{2} - 4 \cdot 1 \cdot (- 2) = 26

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

使用法则

- (- a) = a

= 4^{2} + 4 \cdot 1 \cdot 2

数字相乘:

4 \cdot 1 \cdot 2 = 8

= 4^{2} + 8

4^{2} = 16

= 16 + 8

数字相加:

16 + 8 = 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:

n ab = n a n b

= 2^{2} 2 \cdot 3

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 2 \cdot 3

整理后得

= 26

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

将解分隔开

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

u = \frac{- 4 + 2 6}{2 \cdot 1} : - 2 + 6

\frac{- 4 + 2 6}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

分解

- 4 + 26 : 2 (- 2 + 6)

- 4 + 26

改写为

= - 2 \cdot 2 + 26

因式分解出通项

2

= 2 (- 2 + 6)

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

数字相除:

\frac{2}{2} = 1

= - 2 + 6

u = \frac{- 4 - 2 6}{2 \cdot 1} : - 2 - 6

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

数字相乘:

2 \cdot 1 = 2

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

分解

- 4 - 26 : - 2 (2 + 6)

- 4 - 26

改写为

= - 2 \cdot 2 - 26

因式分解出通项

2

= - 2 (2 + 6)

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

数字相除:

\frac{2}{2} = 1

= - (2 + 6)

取负

- (2 + 6) = - 2 - 6

= - 2 - 6

二次方程组的解是:

u = - 2 + 6, u = - 2 - 6

u = cos (x)

代回

cos (x) = - 2 + 6, cos (x) = - 2 - 6

cos (x) = - 2 + 6, cos (x) = - 2 - 6

cos (x) = - 2 + 6 : x = arccos (- 2 + 6) + 2 πn, x = 2 π - arccos (- 2 + 6) + 2 πn

cos (x) = - 2 + 6

使用反三角函数性质

cos (x) = - 2 + 6

cos (x) = - 2 + 6

的通解

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

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

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

cos (x) = - 2 - 6 :

无解

cos (x) = - 2 - 6

- 1 \leq cos (x) \leq 1

无解

合并所有解

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

以小数形式表示解

x = 1.10460 \dots + 2 πn, x = 2 π - 1.10460 \dots + 2 πn

作图

流行的例子

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sec^{2} (θ) + csc^{2} (θ) = 4

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cos (θ) = 0.56

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