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受欢迎的三角函数 >

2cos^2(x)-7cos(x)+3=0,(0,2pi)

解答

2 cos^{2} (x) - 7 cos (x) + 3 = 0, (0, 2 π)

解答

x = \frac{π}{3}, x = \frac{5 π}{3}

+1

度数

x = 6 0^{\circ}, x = 30 0^{\circ}

求解步骤

2 cos^{2} (x) - 7 cos (x) + 3 = 0, (0, 2 π)

用替代法求解

2 cos^{2} (x) - 7 cos (x) + 3 = 0

令

: cos (x) = u

2 u^{2} - 7 u + 3 = 0

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

2 u^{2} - 7 u + 3 = 0

使用求根公式求解

2 u^{2} - 7 u + 3 = 0

二次方程求根公式:

若

a = 2, b = - 7, c = 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 \cdot 3 = 5

(- 7)^{2} - 4 \cdot 2 \cdot 3

使用指数法则:

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

若

n

是偶数

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 2 \cdot 3

数字相乘:

4 \cdot 2 \cdot 3 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

数字相减:

49 - 24 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 7 ) + 5}{2 \cdot 2}, u_{2} = \frac{- ( - 7 ) - 5}{2 \cdot 2}

u = \frac{- ( - 7 ) + 5}{2 \cdot 2} : 3

\frac{- ( - 7 ) + 5}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{7 + 5}{2 \cdot 2}

数字相加:

7 + 5 = 12

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

数字相乘:

2 \cdot 2 = 4

= \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 7 ) - 5}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 7 ) - 5}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{7 - 5}{2 \cdot 2}

数字相减:

7 - 5 = 2

= \frac{2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

约分:

2

= \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

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

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

cos (x) = 3, (0, 2 π) :

无解

cos (x) = 3, (0, 2 π)

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{1}{2}, (0, 2 π) : x = \frac{π}{3}, x = \frac{5 π}{3}

cos (x) = \frac{1}{2}, (0, 2 π)

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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

在

(0, 2 π)

范围内的解

x = \frac{π}{3}, x = \frac{5 π}{3}

合并所有解

x = \frac{π}{3}, x = \frac{5 π}{3}

作图

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