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受欢迎的三角函数 >

3cos^2(x)-8cos(x)+4=0,0<= x<= 180

解答

3 cos^{2} (x) - 8 cos (x) + 4 = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

解答

x = 0.84106 \dots

求解步骤

3 cos^{2} (x) - 8 cos (x) + 4 = 0, 0^{\circ} \leq x \leq 18 0^{\circ}

用替代法求解

3 cos^{2} (x) - 8 cos (x) + 4 = 0

令

: cos (x) = u

3 u^{2} - 8 u + 4 = 0

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

3 u^{2} - 8 u + 4 = 0

使用求根公式求解

3 u^{2} - 8 u + 4 = 0

二次方程求根公式:

若

a = 3, b = - 8, c = 4

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

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

(- 8)^{2} - 4 \cdot 3 \cdot 4 = 4

(- 8)^{2} - 4 \cdot 3 \cdot 4

使用指数法则:

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

若

n

是偶数

(- 8)^{2} = 8^{2}

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

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

数字相减:

64 - 48 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

u_{1, 2} = \frac{- ( - 8 ) \pm 4}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 8 ) + 4}{2 \cdot 3}, u_{2} = \frac{- ( - 8 ) - 4}{2 \cdot 3}

u = \frac{- ( - 8 ) + 4}{2 \cdot 3} : 2

\frac{- ( - 8 ) + 4}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 + 4}{2 \cdot 3}

数字相加:

8 + 4 = 12

= \frac{12}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{12}{6}

数字相除:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 8 ) - 4}{2 \cdot 3} : \frac{2}{3}

\frac{- ( - 8 ) - 4}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{8 - 4}{2 \cdot 3}

数字相减:

8 - 4 = 4

= \frac{4}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{4}{6}

约分:

2

= \frac{2}{3}

二次方程组的解是:

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

u = cos (x)

代回

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

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

cos (x) = 2, 0 \leq x \leq 18 0^{\circ} :

无解

cos (x) = 2, 0 \leq x \leq 18 0^{\circ}

- 1 \leq cos (x) \leq 1

无解

cos (x) = \frac{2}{3}, 0 \leq x \leq 18 0^{\circ} : x = arccos (\frac{2}{3})

cos (x) = \frac{2}{3}, 0 \leq x \leq 18 0^{\circ}

使用反三角函数性质

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

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

的通解

cos (x) = a \Rightarrow x = arccos (a) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (a) + 36 0^{\circ} n

x = arccos (\frac{2}{3}) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n

x = arccos (\frac{2}{3}) + 36 0^{\circ} n, x = 36 0^{\circ} - arccos (\frac{2}{3}) + 36 0^{\circ} n

在

0 \leq x \leq 18 0^{\circ}

范围内的解

x = arccos (\frac{2}{3})

合并所有解

x = arccos (\frac{2}{3})

以小数形式表示解

x = 0.84106 \dots

作图

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

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