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受欢迎的三角函数 >

75cos^2(x)-34cos(x)+3=0

解答

75 cos^{2} (x) - 34 cos (x) + 3 = 0

解答

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = 1.45050 \dots + 2 πn, x = 2 π - 1.45050 \dots + 2 πn

+1

度数

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n, x = 83.10789 \dots^{\circ} + 36 0^{\circ} n, x = 276.89210 \dots^{\circ} + 36 0^{\circ} n

求解步骤

75 cos^{2} (x) - 34 cos (x) + 3 = 0

用替代法求解

75 cos^{2} (x) - 34 cos (x) + 3 = 0

令

: cos (x) = u

75 u^{2} - 34 u + 3 = 0

75 u^{2} - 34 u + 3 = 0 : u = \frac{1}{3}, u = \frac{3}{25}

75 u^{2} - 34 u + 3 = 0

使用求根公式求解

75 u^{2} - 34 u + 3 = 0

二次方程求根公式:

若

a = 75, b = - 34, c = 3

u_{1, 2} = \frac{- ( - 34 ) \pm ( - 34 ) ^{2} - 4 \cdot 75 \cdot 3}{2 \cdot 75}

u_{1, 2} = \frac{- ( - 34 ) \pm ( - 34 ) ^{2} - 4 \cdot 75 \cdot 3}{2 \cdot 75}

(- 34)^{2} - 4 \cdot 75 \cdot 3 = 16

(- 34)^{2} - 4 \cdot 75 \cdot 3

使用指数法则:

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

若

n

是偶数

(- 34)^{2} = 3 4^{2}

= 3 4^{2} - 4 \cdot 75 \cdot 3

数字相乘:

4 \cdot 75 \cdot 3 = 900

= 3 4^{2} - 900

3 4^{2} = 1156

= 1156 - 900

数字相减:

1156 - 900 = 256

= 256

因式分解数字:

256 = 1 6^{2}

= 1 6^{2}

使用根式运算法则:

n a^{n} = a

1 6^{2} = 16

= 16

u_{1, 2} = \frac{- ( - 34 ) \pm 16}{2 \cdot 75}

将解分隔开

u_{1} = \frac{- ( - 34 ) + 16}{2 \cdot 75}, u_{2} = \frac{- ( - 34 ) - 16}{2 \cdot 75}

u = \frac{- ( - 34 ) + 16}{2 \cdot 75} : \frac{1}{3}

\frac{- ( - 34 ) + 16}{2 \cdot 75}

使用法则

- (- a) = a

= \frac{34 + 16}{2 \cdot 75}

数字相加:

34 + 16 = 50

= \frac{50}{2 \cdot 75}

数字相乘:

2 \cdot 75 = 150

= \frac{50}{150}

约分:

50

= \frac{1}{3}

u = \frac{- ( - 34 ) - 16}{2 \cdot 75} : \frac{3}{25}

\frac{- ( - 34 ) - 16}{2 \cdot 75}

使用法则

- (- a) = a

= \frac{34 - 16}{2 \cdot 75}

数字相减:

34 - 16 = 18

= \frac{18}{2 \cdot 75}

数字相乘:

2 \cdot 75 = 150

= \frac{18}{150}

约分:

6

= \frac{3}{25}

二次方程组的解是:

u = \frac{1}{3}, u = \frac{3}{25}

u = cos (x)

代回

cos (x) = \frac{1}{3}, cos (x) = \frac{3}{25}

cos (x) = \frac{1}{3}, cos (x) = \frac{3}{25}

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

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

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

cos (x) = \frac{3}{25} : x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

cos (x) = \frac{3}{25}

使用反三角函数性质

cos (x) = \frac{3}{25}

cos (x) = \frac{3}{25}

的通解

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

x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

合并所有解

x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn, x = arccos (\frac{3}{25}) + 2 πn, x = 2 π - arccos (\frac{3}{25}) + 2 πn

以小数形式表示解

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn, x = 1.45050 \dots + 2 πn, x = 2 π - 1.45050 \dots + 2 πn

作图

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