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受欢迎的三角函数 >

cos^2(x)-9cos(x)-1=0

解答

cos^{2} (x) - 9 cos (x) - 1 = 0

解答

x = 1.68079 \dots + 2 πn, x = - 1.68079 \dots + 2 πn

+1

度数

x = 96.30218 \dots^{\circ} + 36 0^{\circ} n, x = - 96.30218 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos^{2} (x) - 9 cos (x) - 1 = 0

用替代法求解

cos^{2} (x) - 9 cos (x) - 1 = 0

令

: cos (x) = u

u^{2} - 9 u - 1 = 0

u^{2} - 9 u - 1 = 0 : u = \frac{9 + 85}{2}, u = \frac{9 - 85}{2}

u^{2} - 9 u - 1 = 0

使用求根公式求解

u^{2} - 9 u - 1 = 0

二次方程求根公式:

若

a = 1, b = - 9, c = - 1

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 9 ) \pm ( - 9 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 9)^{2} - 4 \cdot 1 \cdot (- 1) = 85

(- 9)^{2} - 4 \cdot 1 \cdot (- 1)

使用法则

- (- a) = a

= (- 9)^{2} + 4 \cdot 1 \cdot 1

使用指数法则:

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

若

n

是偶数

(- 9)^{2} = 9^{2}

= 9^{2} + 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 9^{2} + 4

9^{2} = 81

= 81 + 4

数字相加:

81 + 4 = 85

= 85

u_{1, 2} = \frac{- ( - 9 ) \pm 85}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - 9 ) + 85}{2 \cdot 1}, u_{2} = \frac{- ( - 9 ) - 85}{2 \cdot 1}

u = \frac{- ( - 9 ) + 85}{2 \cdot 1} : \frac{9 + 85}{2}

\frac{- ( - 9 ) + 85}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{9 + 85}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{9 + 85}{2}

u = \frac{- ( - 9 ) - 85}{2 \cdot 1} : \frac{9 - 85}{2}

\frac{- ( - 9 ) - 85}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{9 - 85}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{9 - 85}{2}

二次方程组的解是:

u = \frac{9 + 85}{2}, u = \frac{9 - 85}{2}

u = cos (x)

代回

cos (x) = \frac{9 + 85}{2}, cos (x) = \frac{9 - 85}{2}

cos (x) = \frac{9 + 85}{2}, cos (x) = \frac{9 - 85}{2}

cos (x) = \frac{9 + 85}{2} :

无解

cos (x) = \frac{9 + 85}{2}

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{9 - 85}{2} : x = arccos (\frac{9 - 85}{2}) + 2 πn, x = - arccos (\frac{9 - 85}{2}) + 2 πn

cos (x) = \frac{9 - 85}{2}

使用反三角函数性质

cos (x) = \frac{9 - 85}{2}

cos (x) = \frac{9 - 85}{2}

的通解

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

x = arccos (\frac{9 - 85}{2}) + 2 πn, x = - arccos (\frac{9 - 85}{2}) + 2 πn

x = arccos (\frac{9 - 85}{2}) + 2 πn, x = - arccos (\frac{9 - 85}{2}) + 2 πn

合并所有解

x = arccos (\frac{9 - 85}{2}) + 2 πn, x = - arccos (\frac{9 - 85}{2}) + 2 πn

以小数形式表示解

x = 1.68079 \dots + 2 πn, x = - 1.68079 \dots + 2 πn

作图

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