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受欢迎的三角函数 >

12cos^2(x)+11cos(x)-5=0

解答

12 cos^{2} (x) + 11 cos (x) - 5 = 0

解答

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

+1

度数

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

12 cos^{2} (x) + 11 cos (x) - 5 = 0

用替代法求解

12 cos^{2} (x) + 11 cos (x) - 5 = 0

令

: cos (x) = u

12 u^{2} + 11 u - 5 = 0

12 u^{2} + 11 u - 5 = 0 : u = \frac{1}{3}, u = - \frac{5}{4}

12 u^{2} + 11 u - 5 = 0

使用求根公式求解

12 u^{2} + 11 u - 5 = 0

二次方程求根公式:

若

a = 12, b = 11, c = - 5

u_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 \cdot 12 ( - 5 )}{2 \cdot 12}

u_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 \cdot 12 ( - 5 )}{2 \cdot 12}

1 1^{2} - 4 \cdot 12 (- 5) = 19

1 1^{2} - 4 \cdot 12 (- 5)

使用法则

- (- a) = a

= 1 1^{2} + 4 \cdot 12 \cdot 5

数字相乘:

4 \cdot 12 \cdot 5 = 240

= 1 1^{2} + 240

1 1^{2} = 121

= 121 + 240

数字相加:

121 + 240 = 361

= 361

因式分解数字:

361 = 1 9^{2}

= 1 9^{2}

使用根式运算法则:

n a^{n} = a

1 9^{2} = 19

= 19

u_{1, 2} = \frac{- 11 \pm 19}{2 \cdot 12}

将解分隔开

u_{1} = \frac{- 11 + 19}{2 \cdot 12}, u_{2} = \frac{- 11 - 19}{2 \cdot 12}

u = \frac{- 11 + 19}{2 \cdot 12} : \frac{1}{3}

\frac{- 11 + 19}{2 \cdot 12}

数字相加/相减:

- 11 + 19 = 8

= \frac{8}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{8}{24}

约分:

8

= \frac{1}{3}

u = \frac{- 11 - 19}{2 \cdot 12} : - \frac{5}{4}

\frac{- 11 - 19}{2 \cdot 12}

数字相减:

- 11 - 19 = - 30

= \frac{- 30}{2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{- 30}{24}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{30}{24}

约分:

6

= - \frac{5}{4}

二次方程组的解是:

u = \frac{1}{3}, u = - \frac{5}{4}

u = cos (x)

代回

cos (x) = \frac{1}{3}, cos (x) = - \frac{5}{4}

cos (x) = \frac{1}{3}, cos (x) = - \frac{5}{4}

cos (x) = \frac{1}{3} : x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

cos (x) = \frac{1}{3}

使用反三角函数性质

cos (x) = \frac{1}{3}

cos (x) = \frac{1}{3}

的通解

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

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

cos (x) = - \frac{5}{4} :

无解

cos (x) = - \frac{5}{4}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

以小数形式表示解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

作图

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