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受欢迎的三角函数 >

cos^2(x)-10cos(x)-1=0

解答

cos^{2} (x) - 10 cos (x) - 1 = 0

解答

x = 1.66997 \dots + 2 πn, x = - 1.66997 \dots + 2 πn

+1

度数

x = 95.68271 \dots^{\circ} + 36 0^{\circ} n, x = - 95.68271 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) - 10 cos (x) - 1 = 0

用替代法求解

cos^{2} (x) - 10 cos (x) - 1 = 0

令

: cos (x) = u

u^{2} - 10 u - 1 = 0

u^{2} - 10 u - 1 = 0 : u = 5 + 26, u = 5 - 26

u^{2} - 10 u - 1 = 0

使用求根公式求解

u^{2} - 10 u - 1 = 0

二次方程求根公式:

若

a = 1, b = - 10, c = - 1

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 10 ) \pm ( - 10 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 10)^{2} - 4 \cdot 1 \cdot (- 1) = 226

(- 10)^{2} - 4 \cdot 1 \cdot (- 1)

使用法则

- (- a) = a

= (- 10)^{2} + 4 \cdot 1 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 10)^{2} = 1 0^{2}

= 1 0^{2} + 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 1 0^{2} + 4

1 0^{2} = 100

= 100 + 4

数字相加:

100 + 4 = 104

= 104

104

质因数分解:

2^{3} \cdot 13

104

104

除以

2104 = 52 \cdot 2

= 2 \cdot 52

52

除以

252 = 26 \cdot 2

= 2 \cdot 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 13

2, 13

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 13

= 2^{3} \cdot 13

= 2^{3} \cdot 13

使用指数法则:

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

= 2^{2} \cdot 2 \cdot 13

使用根式运算法则:

n ab = n a n b

= 2^{2} 2 \cdot 13

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 2 \cdot 13

整理后得

= 226

u_{1, 2} = \frac{- ( - 10 ) \pm 2 26}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - 10 ) + 2 26}{2 \cdot 1}, u_{2} = \frac{- ( - 10 ) - 2 26}{2 \cdot 1}

u = \frac{- ( - 10 ) + 2 26}{2 \cdot 1} : 5 + 26

\frac{- ( - 10 ) + 2 26}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{10 + 2 26}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{10 + 2 26}{2}

分解

10 + 226 : 2 (5 + 26)

10 + 226

改写为

= 2 \cdot 5 + 226

因式分解出通项

2

= 2 (5 + 26)

= \frac{2 ( 5 + 26 )}{2}

数字相除:

\frac{2}{2} = 1

= 5 + 26

u = \frac{- ( - 10 ) - 2 26}{2 \cdot 1} : 5 - 26

\frac{- ( - 10 ) - 2 26}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{10 - 2 26}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{10 - 2 26}{2}

分解

10 - 226 : 2 (5 - 26)

10 - 226

改写为

= 2 \cdot 5 - 226

因式分解出通项

2

= 2 (5 - 26)

= \frac{2 ( 5 - 26 )}{2}

数字相除:

\frac{2}{2} = 1

= 5 - 26

二次方程组的解是:

u = 5 + 26, u = 5 - 26

u = cos (x)

代回

cos (x) = 5 + 26, cos (x) = 5 - 26

cos (x) = 5 + 26, cos (x) = 5 - 26

cos (x) = 5 + 26 :

无解

cos (x) = 5 + 26

- 1 \leq cos (x) \leq 1

无解

cos (x) = 5 - 26 : x = arccos (5 - 26) + 2 πn, x = - arccos (5 - 26) + 2 πn

cos (x) = 5 - 26

使用反三角函数性质

cos (x) = 5 - 26

cos (x) = 5 - 26

的通解

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

x = arccos (5 - 26) + 2 πn, x = - arccos (5 - 26) + 2 πn

x = arccos (5 - 26) + 2 πn, x = - arccos (5 - 26) + 2 πn

合并所有解

x = arccos (5 - 26) + 2 πn, x = - arccos (5 - 26) + 2 πn

以小数形式表示解

x = 1.66997 \dots + 2 πn, x = - 1.66997 \dots + 2 πn

作图

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