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受欢迎的三角函数 >

4cos^2(x)-7cos(x)+3=0

解答

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

解答

x = 2 πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 41.40962 \dots^{\circ} + 36 0^{\circ} n, x = 318.59037 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cos (x) = u

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

4 u^{2} - 7 u + 3 = 0 : u = 1, u = \frac{3}{4}

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

使用求根公式求解

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

二次方程求根公式:

若

a = 4, b = - 7, c = 3

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

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

(- 7)^{2} - 4 \cdot 4 \cdot 3 = 1

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

数字相乘:

4 \cdot 4 \cdot 3 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

数字相减:

49 - 48 = 1

= 1

使用法则

1 = 1

= 1

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

将解分隔开

u_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 4}, u_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 4}

u = \frac{- ( - 7 ) + 1}{2 \cdot 4} : 1

\frac{- ( - 7 ) + 1}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{7 + 1}{2 \cdot 4}

数字相加:

7 + 1 = 8

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

数字相乘:

2 \cdot 4 = 8

= \frac{8}{8}

使用法则

\frac{a}{a} = 1

= 1

u = \frac{- ( - 7 ) - 1}{2 \cdot 4} : \frac{3}{4}

\frac{- ( - 7 ) - 1}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{7 - 1}{2 \cdot 4}

数字相减:

7 - 1 = 6

= \frac{6}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{6}{8}

约分:

2

= \frac{3}{4}

二次方程组的解是:

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

u = cos (x)

代回

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

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

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

cos (x) = \frac{3}{4}

使用反三角函数性质

cos (x) = \frac{3}{4}

cos (x) = \frac{3}{4}

的通解

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

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

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

合并所有解

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

以小数形式表示解

x = 2 πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

作图

流行的例子

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