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受欢迎的三角函数 >

8cos(2x)+6=cos^2(x)+cos(x)

解答

8 cos (2 x) + 6 = cos^{2} (x) + cos (x)

解答

x = 1.15927 \dots + 2 πn, x = 2 π - 1.15927 \dots + 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

+1

度数

x = 66.42182 \dots^{\circ} + 36 0^{\circ} n, x = 293.57817 \dots^{\circ} + 36 0^{\circ} n, x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

求解步骤

8 cos (2 x) + 6 = cos^{2} (x) + cos (x)

两边减去

cos^{2} (x) + cos (x)

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

使用三角恒等式改写

6 - cos (x) - cos^{2} (x) + 8 cos (2 x)

使用倍角公式:

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

= 6 - cos (x) - cos^{2} (x) + 8 (2 cos^{2} (x) - 1)

化简

6 - cos (x) - cos^{2} (x) + 8 (2 cos^{2} (x) - 1) : 15 cos^{2} (x) - cos (x) - 2

6 - cos (x) - cos^{2} (x) + 8 (2 cos^{2} (x) - 1)

乘开

8 (2 cos^{2} (x) - 1) : 16 cos^{2} (x) - 8

8 (2 cos^{2} (x) - 1)

使用分配律:

a (b - c) = ab - a c

a = 8, b = 2 cos^{2} (x), c = 1

= 8 \cdot 2 cos^{2} (x) - 8 \cdot 1

化简

8 \cdot 2 cos^{2} (x) - 8 \cdot 1 : 16 cos^{2} (x) - 8

8 \cdot 2 cos^{2} (x) - 8 \cdot 1

数字相乘:

8 \cdot 2 = 16

= 16 cos^{2} (x) - 8 \cdot 1

数字相乘:

8 \cdot 1 = 8

= 16 cos^{2} (x) - 8

= 16 cos^{2} (x) - 8

= 6 - cos (x) - cos^{2} (x) + 16 cos^{2} (x) - 8

化简

6 - cos (x) - cos^{2} (x) + 16 cos^{2} (x) - 8 : 15 cos^{2} (x) - cos (x) - 2

6 - cos (x) - cos^{2} (x) + 16 cos^{2} (x) - 8

同类项相加:

- cos^{2} (x) + 16 cos^{2} (x) = 15 cos^{2} (x)

= 6 - cos (x) + 15 cos^{2} (x) - 8

对同类项分组

= - cos (x) + 15 cos^{2} (x) + 6 - 8

数字相加/相减:

6 - 8 = - 2

= 15 cos^{2} (x) - cos (x) - 2

= 15 cos^{2} (x) - cos (x) - 2

= 15 cos^{2} (x) - cos (x) - 2

- 2 - cos (x) + 15 cos^{2} (x) = 0

用替代法求解

- 2 - cos (x) + 15 cos^{2} (x) = 0

令

: cos (x) = u

- 2 - u + 15 u^{2} = 0

- 2 - u + 15 u^{2} = 0 : u = \frac{2}{5}, u = - \frac{1}{3}

- 2 - u + 15 u^{2} = 0

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 15, b = - 1, c = - 2

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 15 ( - 2 )}{2 \cdot 15}

u_{1, 2} = \frac{- ( - 1 ) \pm ( - 1 ) ^{2} - 4 \cdot 15 ( - 2 )}{2 \cdot 15}

(- 1)^{2} - 4 \cdot 15 (- 2) = 11

(- 1)^{2} - 4 \cdot 15 (- 2)

使用法则

- (- a) = a

= (- 1)^{2} + 4 \cdot 15 \cdot 2

(- 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 15 \cdot 2 = 120

4 \cdot 15 \cdot 2

数字相乘:

4 \cdot 15 \cdot 2 = 120

= 120

= 1 + 120

数字相加:

1 + 120 = 121

= 121

因式分解数字:

121 = 1 1^{2}

= 1 1^{2}

使用根式运算法则:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 1 ) \pm 11}{2 \cdot 15}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 11}{2 \cdot 15}, u_{2} = \frac{- ( - 1 ) - 11}{2 \cdot 15}

u = \frac{- ( - 1 ) + 11}{2 \cdot 15} : \frac{2}{5}

\frac{- ( - 1 ) + 11}{2 \cdot 15}

使用法则

- (- a) = a

= \frac{1 + 11}{2 \cdot 15}

数字相加:

1 + 11 = 12

= \frac{12}{2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{12}{30}

约分:

6

= \frac{2}{5}

u = \frac{- ( - 1 ) - 11}{2 \cdot 15} : - \frac{1}{3}

\frac{- ( - 1 ) - 11}{2 \cdot 15}

使用法则

- (- a) = a

= \frac{1 - 11}{2 \cdot 15}

数字相减:

1 - 11 = - 10

= \frac{- 10}{2 \cdot 15}

数字相乘:

2 \cdot 15 = 30

= \frac{- 10}{30}

使用分式法则:

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

= - \frac{10}{30}

约分:

10

= - \frac{1}{3}

二次方程组的解是:

u = \frac{2}{5}, u = - \frac{1}{3}

u = cos (x)

代回

cos (x) = \frac{2}{5}, cos (x) = - \frac{1}{3}

cos (x) = \frac{2}{5}, cos (x) = - \frac{1}{3}

cos (x) = \frac{2}{5} : x = arccos (\frac{2}{5}) + 2 πn, x = 2 π - arccos (\frac{2}{5}) + 2 πn

cos (x) = \frac{2}{5}

使用反三角函数性质

cos (x) = \frac{2}{5}

cos (x) = \frac{2}{5}

的通解

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

x = arccos (\frac{2}{5}) + 2 πn, x = 2 π - arccos (\frac{2}{5}) + 2 πn

x = arccos (\frac{2}{5}) + 2 πn, x = 2 π - arccos (\frac{2}{5}) + 2 πn

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

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

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

合并所有解

x = arccos (\frac{2}{5}) + 2 πn, x = 2 π - arccos (\frac{2}{5}) + 2 πn, x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

以小数形式表示解

x = 1.15927 \dots + 2 πn, x = 2 π - 1.15927 \dots + 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

作图

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