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受欢迎的三角函数 >

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

解答

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

解答

x \in R 无解

求解步骤

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

用替代法求解

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

令

: cos (x) = u

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

2 u^{2} - u + 1 = 0 : u = \frac{1}{4} + i \frac{7}{4}, u = \frac{1}{4} - i \frac{7}{4}

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

化简

(- 1)^{2} - 4 \cdot 2 \cdot 1 : 7 i

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

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 8

= 1 - 8

数字相减:

1 - 8 = - 7

= - 7

使用根式运算法则:

- a = - 1 a

- 7 = - 1 7

= - 1 7

使用虚数运算法则:

- 1 = i

= 7 i

u_{1, 2} = \frac{- ( - 1 ) \pm 7 i}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 7 i}{2 \cdot 2}, u_{2} = \frac{- ( - 1 ) - 7 i}{2 \cdot 2}

u = \frac{- ( - 1 ) + 7 i}{2 \cdot 2} : \frac{1}{4} + i \frac{7}{4}

\frac{- ( - 1 ) + 7 i}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{1 + 7 i}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{1 + 7 i}{4}

将

\frac{1 + 7 i}{4}

改写成标准复数形式:

\frac{1}{4} + \frac{7}{4} i

\frac{1 + 7 i}{4}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 + 7 i}{4} = \frac{1}{4} + \frac{7 i}{4}

= \frac{1}{4} + \frac{7 i}{4}

= \frac{1}{4} + \frac{7}{4} i

u = \frac{- ( - 1 ) - 7 i}{2 \cdot 2} : \frac{1}{4} - i \frac{7}{4}

\frac{- ( - 1 ) - 7 i}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{1 - 7 i}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{1 - 7 i}{4}

将

\frac{1 - 7 i}{4}

改写成标准复数形式:

\frac{1}{4} - \frac{7}{4} i

\frac{1 - 7 i}{4}

使用分式法则:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 - 7 i}{4} = \frac{1}{4} - \frac{7 i}{4}

= \frac{1}{4} - \frac{7 i}{4}

= \frac{1}{4} - \frac{7}{4} i

二次方程组的解是:

u = \frac{1}{4} + i \frac{7}{4}, u = \frac{1}{4} - i \frac{7}{4}

u = cos (x)

代回

cos (x) = \frac{1}{4} + i \frac{7}{4}, cos (x) = \frac{1}{4} - i \frac{7}{4}

cos (x) = \frac{1}{4} + i \frac{7}{4}, cos (x) = \frac{1}{4} - i \frac{7}{4}

cos (x) = \frac{1}{4} + i \frac{7}{4} :

无解

cos (x) = \frac{1}{4} + i \frac{7}{4}

无解

cos (x) = \frac{1}{4} - i \frac{7}{4} :

无解

cos (x) = \frac{1}{4} - i \frac{7}{4}

无解

合并所有解

x \in R 无解

作图

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