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受欢迎的三角函数 >

3cos(2x)-2cos(x)-1=0

解答

3 cos (2 x) - 2 cos (x) - 1 = 0

解答

x = 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 131.81031 \dots^{\circ} + 36 0^{\circ} n, x = - 131.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

3 cos (2 x) - 2 cos (x) - 1 = 0

使用三角恒等式改写

- 1 - 2 cos (x) + 3 cos (2 x)

使用倍角公式:

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

= - 1 - 2 cos (x) + 3 (2 cos^{2} (x) - 1)

化简

- 1 - 2 cos (x) + 3 (2 cos^{2} (x) - 1) : 6 cos^{2} (x) - 2 cos (x) - 4

- 1 - 2 cos (x) + 3 (2 cos^{2} (x) - 1)

乘开

3 (2 cos^{2} (x) - 1) : 6 cos^{2} (x) - 3

3 (2 cos^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = 3, b = 2 cos^{2} (x), c = 1

= 3 \cdot 2 cos^{2} (x) - 3 \cdot 1

化简

3 \cdot 2 cos^{2} (x) - 3 \cdot 1 : 6 cos^{2} (x) - 3

3 \cdot 2 cos^{2} (x) - 3 \cdot 1

数字相乘:

3 \cdot 2 = 6

= 6 cos^{2} (x) - 3 \cdot 1

数字相乘:

3 \cdot 1 = 3

= 6 cos^{2} (x) - 3

= 6 cos^{2} (x) - 3

= - 1 - 2 cos (x) + 6 cos^{2} (x) - 3

化简

- 1 - 2 cos (x) + 6 cos^{2} (x) - 3 : 6 cos^{2} (x) - 2 cos (x) - 4

- 1 - 2 cos (x) + 6 cos^{2} (x) - 3

对同类项分组

= - 2 cos (x) + 6 cos^{2} (x) - 1 - 3

数字相减:

- 1 - 3 = - 4

= 6 cos^{2} (x) - 2 cos (x) - 4

= 6 cos^{2} (x) - 2 cos (x) - 4

= 6 cos^{2} (x) - 2 cos (x) - 4

- 4 - 2 cos (x) + 6 cos^{2} (x) = 0

用替代法求解

- 4 - 2 cos (x) + 6 cos^{2} (x) = 0

令

: cos (x) = u

- 4 - 2 u + 6 u^{2} = 0

- 4 - 2 u + 6 u^{2} = 0 : u = 1, u = - \frac{2}{3}

- 4 - 2 u + 6 u^{2} = 0

改写成标准形式

a x^{2} + b x + c = 0

6 u^{2} - 2 u - 4 = 0

使用求根公式求解

6 u^{2} - 2 u - 4 = 0

二次方程求根公式:

若

a = 6, b = - 2, c = - 4

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 6 ( - 4 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 6 ( - 4 )}{2 \cdot 6}

(- 2)^{2} - 4 \cdot 6 (- 4) = 10

(- 2)^{2} - 4 \cdot 6 (- 4)

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 6 \cdot 4

使用指数法则:

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

若

n

是偶数

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

= 2^{2} + 4 \cdot 6 \cdot 4

数字相乘:

4 \cdot 6 \cdot 4 = 96

= 2^{2} + 96

2^{2} = 4

= 4 + 96

数字相加:

4 + 96 = 100

= 100

因式分解数字:

100 = 1 0^{2}

= 1 0^{2}

使用根式运算法则:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- ( - 2 ) \pm 10}{2 \cdot 6}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 10}{2 \cdot 6}, u_{2} = \frac{- ( - 2 ) - 10}{2 \cdot 6}

u = \frac{- ( - 2 ) + 10}{2 \cdot 6} : 1

\frac{- ( - 2 ) + 10}{2 \cdot 6}

使用法则

- (- a) = a

= \frac{2 + 10}{2 \cdot 6}

数字相加:

2 + 10 = 12

= \frac{12}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{12}{12}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- ( - 2 ) - 10}{2 \cdot 6} : - \frac{2}{3}

\frac{- ( - 2 ) - 10}{2 \cdot 6}

使用法则

- (- a) = a

= \frac{2 - 10}{2 \cdot 6}

数字相减:

2 - 10 = - 8

= \frac{- 8}{2 \cdot 6}

数字相乘:

2 \cdot 6 = 12

= \frac{- 8}{12}

使用分式法则:

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

= - \frac{8}{12}

约分:

4

= - \frac{2}{3}

二次方程组的解是:

u = 1, u = - \frac{2}{3}

u = cos (x)

代回

cos (x) = 1, cos (x) = - \frac{2}{3}

cos (x) = 1, cos (x) = - \frac{2}{3}

cos (x) = 1 : x = 2 πn

cos (x) = 1

cos (x) = 1

的通解

cos (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

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

cos (x) = - \frac{2}{3}

使用反三角函数性质

cos (x) = - \frac{2}{3}

cos (x) = - \frac{2}{3}

的通解

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

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

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

合并所有解

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

以小数形式表示解

x = 2 πn, x = 2.30052 \dots + 2 πn, x = - 2.30052 \dots + 2 πn

作图

流行的例子

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