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受欢迎的三角函数 >

10cos^2(x)+11cos(x)+3=0

解答

10 cos^{2} (x) + 11 cos (x) + 3 = 0

解答

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = - 126.86989 \dots^{\circ} + 36 0^{\circ} n

求解步骤

10 cos^{2} (x) + 11 cos (x) + 3 = 0

用替代法求解

10 cos^{2} (x) + 11 cos (x) + 3 = 0

令

: cos (x) = u

10 u^{2} + 11 u + 3 = 0

10 u^{2} + 11 u + 3 = 0 : u = - \frac{1}{2}, u = - \frac{3}{5}

10 u^{2} + 11 u + 3 = 0

使用求根公式求解

10 u^{2} + 11 u + 3 = 0

二次方程求根公式:

若

a = 10, b = 11, c = 3

u_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 \cdot 10 \cdot 3}{2 \cdot 10}

u_{1, 2} = \frac{- 11 \pm 1 1 ^{2} - 4 \cdot 10 \cdot 3}{2 \cdot 10}

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

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

数字相乘:

4 \cdot 10 \cdot 3 = 120

= 1 1^{2} - 120

1 1^{2} = 121

= 121 - 120

数字相减:

121 - 120 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 11 \pm 1}{2 \cdot 10}

将解分隔开

u_{1} = \frac{- 11 + 1}{2 \cdot 10}, u_{2} = \frac{- 11 - 1}{2 \cdot 10}

u = \frac{- 11 + 1}{2 \cdot 10} : - \frac{1}{2}

\frac{- 11 + 1}{2 \cdot 10}

数字相加/相减:

- 11 + 1 = - 10

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

数字相乘:

2 \cdot 10 = 20

= \frac{- 10}{20}

使用分式法则:

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

= - \frac{10}{20}

约分:

10

= - \frac{1}{2}

u = \frac{- 11 - 1}{2 \cdot 10} : - \frac{3}{5}

\frac{- 11 - 1}{2 \cdot 10}

数字相减:

- 11 - 1 = - 12

= \frac{- 12}{2 \cdot 10}

数字相乘:

2 \cdot 10 = 20

= \frac{- 12}{20}

使用分式法则:

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

= - \frac{12}{20}

约分:

4

= - \frac{3}{5}

二次方程组的解是:

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

u = cos (x)

代回

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

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

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 = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

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

使用反三角函数性质

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

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

的通解

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

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

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

合并所有解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

以小数形式表示解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

作图

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