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受欢迎的三角函数 >

cos^2(x)-8cos(x)-1=0

解答

cos^{2} (x) - 8 cos (x) - 1 = 0

解答

x = 1.69421 \dots + 2 πn, x = - 1.69421 \dots + 2 πn

+1

度数

x = 97.07137 \dots^{\circ} + 36 0^{\circ} n, x = - 97.07137 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) - 8 cos (x) - 1 = 0

用替代法求解

cos^{2} (x) - 8 cos (x) - 1 = 0

令

: cos (x) = u

u^{2} - 8 u - 1 = 0

u^{2} - 8 u - 1 = 0 : u = 4 + 17, u = 4 - 17

u^{2} - 8 u - 1 = 0

使用求根公式求解

u^{2} - 8 u - 1 = 0

二次方程求根公式:

若

a = 1, b = - 8, c = - 1

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 8)^{2} - 4 \cdot 1 \cdot (- 1) = 217

(- 8)^{2} - 4 \cdot 1 \cdot (- 1)

使用法则

- (- a) = a

= (- 8)^{2} + 4 \cdot 1 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 8)^{2} = 8^{2}

= 8^{2} + 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 8^{2} + 4

8^{2} = 64

= 64 + 4

数字相加:

64 + 4 = 68

= 68

68

质因数分解:

2^{2} \cdot 17

68

68

除以

268 = 34 \cdot 2

= 2 \cdot 34

34

除以

234 = 17 \cdot 2

= 2 \cdot 2 \cdot 17

2, 17

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 17

= 2^{2} \cdot 17

= 2^{2} \cdot 17

使用根式运算法则:

n ab = n a n b

= 17 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 217

u_{1, 2} = \frac{- ( - 8 ) \pm 2 17}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - 8 ) + 2 17}{2 \cdot 1}, u_{2} = \frac{- ( - 8 ) - 2 17}{2 \cdot 1}

u = \frac{- ( - 8 ) + 2 17}{2 \cdot 1} : 4 + 17

\frac{- ( - 8 ) + 2 17}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{8 + 2 17}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8 + 2 17}{2}

分解

8 + 217 : 2 (4 + 17)

8 + 217

改写为

= 2 \cdot 4 + 217

因式分解出通项

2

= 2 (4 + 17)

= \frac{2 ( 4 + 17 )}{2}

数字相除:

\frac{2}{2} = 1

= 4 + 17

u = \frac{- ( - 8 ) - 2 17}{2 \cdot 1} : 4 - 17

\frac{- ( - 8 ) - 2 17}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{8 - 2 17}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8 - 2 17}{2}

分解

8 - 217 : 2 (4 - 17)

8 - 217

改写为

= 2 \cdot 4 - 217

因式分解出通项

2

= 2 (4 - 17)

= \frac{2 ( 4 - 17 )}{2}

数字相除:

\frac{2}{2} = 1

= 4 - 17

二次方程组的解是:

u = 4 + 17, u = 4 - 17

u = cos (x)

代回

cos (x) = 4 + 17, cos (x) = 4 - 17

cos (x) = 4 + 17, cos (x) = 4 - 17

cos (x) = 4 + 17 :

无解

cos (x) = 4 + 17

- 1 \leq cos (x) \leq 1

无解

cos (x) = 4 - 17 : x = arccos (4 - 17) + 2 πn, x = - arccos (4 - 17) + 2 πn

cos (x) = 4 - 17

使用反三角函数性质

cos (x) = 4 - 17

cos (x) = 4 - 17

的通解

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

x = arccos (4 - 17) + 2 πn, x = - arccos (4 - 17) + 2 πn

x = arccos (4 - 17) + 2 πn, x = - arccos (4 - 17) + 2 πn

合并所有解

x = arccos (4 - 17) + 2 πn, x = - arccos (4 - 17) + 2 πn

以小数形式表示解

x = 1.69421 \dots + 2 πn, x = - 1.69421 \dots + 2 πn

作图

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