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受欢迎的三角函数 >

4cos^2(x)-5cos(x)-6=0

解答

4 cos^{2} (x) - 5 cos (x) - 6 = 0

解答

x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

+1

度数

x = 138.59037 \dots^{\circ} + 36 0^{\circ} n, x = - 138.59037 \dots^{\circ} + 36 0^{\circ} n

求解步骤

4 cos^{2} (x) - 5 cos (x) - 6 = 0

用替代法求解

4 cos^{2} (x) - 5 cos (x) - 6 = 0

令

: cos (x) = u

4 u^{2} - 5 u - 6 = 0

4 u^{2} - 5 u - 6 = 0 : u = 2, u = - \frac{3}{4}

4 u^{2} - 5 u - 6 = 0

使用求根公式求解

4 u^{2} - 5 u - 6 = 0

二次方程求根公式:

若

a = 4, b = - 5, c = - 6

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 4 ( - 6 )}{2 \cdot 4}

(- 5)^{2} - 4 \cdot 4 (- 6) = 11

(- 5)^{2} - 4 \cdot 4 (- 6)

使用法则

- (- a) = a

= (- 5)^{2} + 4 \cdot 4 \cdot 6

使用指数法则:

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

若

n

是偶数

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 4 \cdot 6

数字相乘:

4 \cdot 4 \cdot 6 = 96

= 5^{2} + 96

5^{2} = 25

= 25 + 96

数字相加:

25 + 96 = 121

= 121

因式分解数字:

121 = 1 1^{2}

= 1 1^{2}

使用根式运算法则:

n a^{n} = a

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- ( - 5 ) \pm 11}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 11}{2 \cdot 4}, u_{2} = \frac{- ( - 5 ) - 11}{2 \cdot 4}

u = \frac{- ( - 5 ) + 11}{2 \cdot 4} : 2

\frac{- ( - 5 ) + 11}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{5 + 11}{2 \cdot 4}

数字相加:

5 + 11 = 16

= \frac{16}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{16}{8}

数字相除:

\frac{16}{8} = 2

= 2

u = \frac{- ( - 5 ) - 11}{2 \cdot 4} : - \frac{3}{4}

\frac{- ( - 5 ) - 11}{2 \cdot 4}

使用法则

- (- a) = a

= \frac{5 - 11}{2 \cdot 4}

数字相减:

5 - 11 = - 6

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 6}{8}

使用分式法则:

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

= - \frac{6}{8}

约分:

2

= - \frac{3}{4}

二次方程组的解是:

u = 2, u = - \frac{3}{4}

u = cos (x)

代回

cos (x) = 2, cos (x) = - \frac{3}{4}

cos (x) = 2, cos (x) = - \frac{3}{4}

cos (x) = 2 :

无解

cos (x) = 2

- 1 \leq cos (x) \leq 1

无解

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

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

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

合并所有解

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

以小数形式表示解

x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

作图

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