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受欢迎的三角函数 >

3cos^2(x)-2cos(x)-2=0,0<= x<2pi

解答

3 cos^{2} (x) - 2 cos (x) - 2 = 0, 0 \leq x < 2 π

解答

x = 2.15146 \dots, x = - 2.15146 \dots + 2 π

+1

度数

x = 123.26990 \dots^{\circ}, x = 236.73009 \dots^{\circ}

求解步骤

3 cos^{2} (x) - 2 cos (x) - 2 = 0, 0 \leq x < 2 π

用替代法求解

3 cos^{2} (x) - 2 cos (x) - 2 = 0

令

: cos (x) = u

3 u^{2} - 2 u - 2 = 0

3 u^{2} - 2 u - 2 = 0 : u = \frac{1 + 7}{3}, u = \frac{1 - 7}{3}

3 u^{2} - 2 u - 2 = 0

使用求根公式求解

3 u^{2} - 2 u - 2 = 0

二次方程求根公式:

若

a = 3, b = - 2, c = - 2

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 3 ( - 2 )}{2 \cdot 3}

(- 2)^{2} - 4 \cdot 3 (- 2) = 27

(- 2)^{2} - 4 \cdot 3 (- 2)

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 3 \cdot 2

使用指数法则:

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

若

n

是偶数

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

= 2^{2} + 4 \cdot 3 \cdot 2

数字相乘:

4 \cdot 3 \cdot 2 = 24

= 2^{2} + 24

2^{2} = 4

= 4 + 24

数字相加:

4 + 24 = 28

= 28

28

质因数分解:

2^{2} \cdot 7

28

28

除以

228 = 14 \cdot 2

= 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 7

= 2^{2} \cdot 7

= 2^{2} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 27

u_{1, 2} = \frac{- ( - 2 ) \pm 2 7}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 2 7}{2 \cdot 3}, u_{2} = \frac{- ( - 2 ) - 2 7}{2 \cdot 3}

u = \frac{- ( - 2 ) + 2 7}{2 \cdot 3} : \frac{1 + 7}{3}

\frac{- ( - 2 ) + 2 7}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{2 + 2 7}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{2 + 2 7}{6}

分解

2 + 27 : 2 (1 + 7)

2 + 27

改写为

= 2 \cdot 1 + 27

因式分解出通项

2

= 2 (1 + 7)

= \frac{2 ( 1 + 7 )}{6}

约分:

2

= \frac{1 + 7}{3}

u = \frac{- ( - 2 ) - 2 7}{2 \cdot 3} : \frac{1 - 7}{3}

\frac{- ( - 2 ) - 2 7}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{2 - 2 7}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{2 - 2 7}{6}

分解

2 - 27 : 2 (1 - 7)

2 - 27

改写为

= 2 \cdot 1 - 27

因式分解出通项

2

= 2 (1 - 7)

= \frac{2 ( 1 - 7 )}{6}

约分:

2

= \frac{1 - 7}{3}

二次方程组的解是:

u = \frac{1 + 7}{3}, u = \frac{1 - 7}{3}

u = cos (x)

代回

cos (x) = \frac{1 + 7}{3}, cos (x) = \frac{1 - 7}{3}

cos (x) = \frac{1 + 7}{3}, cos (x) = \frac{1 - 7}{3}

cos (x) = \frac{1 + 7}{3}, 0 \leq x < 2 π :

无解

cos (x) = \frac{1 + 7}{3}, 0 \leq x < 2 π

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{1 - 7}{3}, 0 \leq x < 2 π : x = arccos (\frac{1 - 7}{3}), x = - arccos (\frac{1 - 7}{3}) + 2 π

cos (x) = \frac{1 - 7}{3}, 0 \leq x < 2 π

使用反三角函数性质

cos (x) = \frac{1 - 7}{3}

cos (x) = \frac{1 - 7}{3}

的通解

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

x = arccos (\frac{1 - 7}{3}) + 2 πn, x = - arccos (\frac{1 - 7}{3}) + 2 πn

x = arccos (\frac{1 - 7}{3}) + 2 πn, x = - arccos (\frac{1 - 7}{3}) + 2 πn

在

0 \leq x < 2 π

范围内的解

x = arccos (\frac{1 - 7}{3}), x = - arccos (\frac{1 - 7}{3}) + 2 π

合并所有解

x = arccos (\frac{1 - 7}{3}), x = - arccos (\frac{1 - 7}{3}) + 2 π

以小数形式表示解

x = 2.15146 \dots, x = - 2.15146 \dots + 2 π

作图

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