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受欢迎的三角函数 >

2cos^2(x)+9cos(x)=4

解答

2 cos^{2} (x) + 9 cos (x) = 4

解答

x = 1.15104 \dots + 2 πn, x = 2 π - 1.15104 \dots + 2 πn

+1

度数

x = 65.94982 \dots^{\circ} + 36 0^{\circ} n, x = 294.05017 \dots^{\circ} + 36 0^{\circ} n

求解步骤

2 cos^{2} (x) + 9 cos (x) = 4

用替代法求解

2 cos^{2} (x) + 9 cos (x) = 4

令

: cos (x) = u

2 u^{2} + 9 u = 4

2 u^{2} + 9 u = 4 : u = \frac{- 9 + 113}{4}, u = \frac{- 9 - 113}{4}

2 u^{2} + 9 u = 4

将

4

para o lado esquerdo

2 u^{2} + 9 u = 4

两边减去

4

2 u^{2} + 9 u - 4 = 4 - 4

化简

2 u^{2} + 9 u - 4 = 0

2 u^{2} + 9 u - 4 = 0

使用求根公式求解

2 u^{2} + 9 u - 4 = 0

二次方程求根公式:

若

a = 2, b = 9, c = - 4

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

u_{1, 2} = \frac{- 9 \pm 9 ^{2} - 4 \cdot 2 ( - 4 )}{2 \cdot 2}

9^{2} - 4 \cdot 2 (- 4) = 113

9^{2} - 4 \cdot 2 (- 4)

使用法则

- (- a) = a

= 9^{2} + 4 \cdot 2 \cdot 4

数字相乘:

4 \cdot 2 \cdot 4 = 32

= 9^{2} + 32

9^{2} = 81

= 81 + 32

数字相加:

81 + 32 = 113

= 113

u_{1, 2} = \frac{- 9 \pm 113}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 9 + 113}{2 \cdot 2}, u_{2} = \frac{- 9 - 113}{2 \cdot 2}

u = \frac{- 9 + 113}{2 \cdot 2} : \frac{- 9 + 113}{4}

\frac{- 9 + 113}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 9 + 113}{4}

u = \frac{- 9 - 113}{2 \cdot 2} : \frac{- 9 - 113}{4}

\frac{- 9 - 113}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 9 - 113}{4}

二次方程组的解是:

u = \frac{- 9 + 113}{4}, u = \frac{- 9 - 113}{4}

u = cos (x)

代回

cos (x) = \frac{- 9 + 113}{4}, cos (x) = \frac{- 9 - 113}{4}

cos (x) = \frac{- 9 + 113}{4}, cos (x) = \frac{- 9 - 113}{4}

cos (x) = \frac{- 9 + 113}{4} : x = arccos (\frac{- 9 + 113}{4}) + 2 πn, x = 2 π - arccos (\frac{- 9 + 113}{4}) + 2 πn

cos (x) = \frac{- 9 + 113}{4}

使用反三角函数性质

cos (x) = \frac{- 9 + 113}{4}

cos (x) = \frac{- 9 + 113}{4}

的通解

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

x = arccos (\frac{- 9 + 113}{4}) + 2 πn, x = 2 π - arccos (\frac{- 9 + 113}{4}) + 2 πn

x = arccos (\frac{- 9 + 113}{4}) + 2 πn, x = 2 π - arccos (\frac{- 9 + 113}{4}) + 2 πn

cos (x) = \frac{- 9 - 113}{4} :

无解

cos (x) = \frac{- 9 - 113}{4}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{- 9 + 113}{4}) + 2 πn, x = 2 π - arccos (\frac{- 9 + 113}{4}) + 2 πn

以小数形式表示解

x = 1.15104 \dots + 2 πn, x = 2 π - 1.15104 \dots + 2 πn

作图

流行的例子

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