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受欢迎的三角函数 >

8cos^2(x)+14cos(x)+3=0

解答

8 cos^{2} (x) + 14 cos (x) + 3 = 0

解答

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

+1

度数

x = 104.47751 \dots^{\circ} + 36 0^{\circ} n, x = - 104.47751 \dots^{\circ} + 36 0^{\circ} n

求解步骤

8 cos^{2} (x) + 14 cos (x) + 3 = 0

用替代法求解

8 cos^{2} (x) + 14 cos (x) + 3 = 0

令

: cos (x) = u

8 u^{2} + 14 u + 3 = 0

8 u^{2} + 14 u + 3 = 0 : u = - \frac{1}{4}, u = - \frac{3}{2}

8 u^{2} + 14 u + 3 = 0

使用求根公式求解

8 u^{2} + 14 u + 3 = 0

二次方程求根公式:

若

a = 8, b = 14, c = 3

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 \cdot 8 \cdot 3}{2 \cdot 8}

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 \cdot 8 \cdot 3}{2 \cdot 8}

1 4^{2} - 4 \cdot 8 \cdot 3 = 10

1 4^{2} - 4 \cdot 8 \cdot 3

数字相乘:

4 \cdot 8 \cdot 3 = 96

= 1 4^{2} - 96

1 4^{2} = 196

= 196 - 96

数字相减:

196 - 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{- 14 \pm 10}{2 \cdot 8}

将解分隔开

u_{1} = \frac{- 14 + 10}{2 \cdot 8}, u_{2} = \frac{- 14 - 10}{2 \cdot 8}

u = \frac{- 14 + 10}{2 \cdot 8} : - \frac{1}{4}

\frac{- 14 + 10}{2 \cdot 8}

数字相加/相减:

- 14 + 10 = - 4

= \frac{- 4}{2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 4}{16}

使用分式法则:

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

= - \frac{4}{16}

约分:

4

= - \frac{1}{4}

u = \frac{- 14 - 10}{2 \cdot 8} : - \frac{3}{2}

\frac{- 14 - 10}{2 \cdot 8}

数字相减:

- 14 - 10 = - 24

= \frac{- 24}{2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 24}{16}

使用分式法则:

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

= - \frac{24}{16}

约分:

8

= - \frac{3}{2}

二次方程组的解是:

u = - \frac{1}{4}, u = - \frac{3}{2}

u = cos (x)

代回

cos (x) = - \frac{1}{4}, cos (x) = - \frac{3}{2}

cos (x) = - \frac{1}{4}, cos (x) = - \frac{3}{2}

cos (x) = - \frac{1}{4} : x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

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

使用反三角函数性质

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

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

的通解

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

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

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

无解

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

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (- \frac{1}{4}) + 2 πn, x = - arccos (- \frac{1}{4}) + 2 πn

以小数形式表示解

x = 1.82347 \dots + 2 πn, x = - 1.82347 \dots + 2 πn

作图

流行的例子

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cos(x)=(sqrt(2))/4

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