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受欢迎的三角函数 >

cos(2x)+5cos(x)=3

解答

cos (2 x) + 5 cos (x) = 3

解答

x = 0.87960 \dots + 2 πn, x = 2 π - 0.87960 \dots + 2 πn

+1

度数

x = 50.39742 \dots^{\circ} + 36 0^{\circ} n, x = 309.60257 \dots^{\circ} + 36 0^{\circ} n

求解步骤

cos (2 x) + 5 cos (x) = 3

两边减去

3

cos (2 x) + 5 cos (x) - 3 = 0

使用三角恒等式改写

- 3 + cos (2 x) + 5 cos (x)

使用倍角公式:

cos (2 x) = 2 cos^{2} (x) - 1

= - 3 + 2 cos^{2} (x) - 1 + 5 cos (x)

化简

- 3 + 2 cos^{2} (x) - 1 + 5 cos (x) : 2 cos^{2} (x) + 5 cos (x) - 4

- 3 + 2 cos^{2} (x) - 1 + 5 cos (x)

对同类项分组

= 2 cos^{2} (x) + 5 cos (x) - 3 - 1

数字相减:

- 3 - 1 = - 4

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

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

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

用替代法求解

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

令

: cos (x) = u

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

- 4 + 2 u^{2} + 5 u = 0 : u = \frac{- 5 + 57}{4}, u = \frac{- 5 - 57}{4}

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

改写成标准形式

a x^{2} + b x + c = 0

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

使用求根公式求解

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

二次方程求根公式:

若

a = 2, b = 5, c = - 4

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

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

5^{2} - 4 \cdot 2 (- 4) = 57

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 2 \cdot 4 = 32

= 5^{2} + 32

5^{2} = 25

= 25 + 32

数字相加:

25 + 32 = 57

= 57

u_{1, 2} = \frac{- 5 \pm 57}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 5 + 57}{2 \cdot 2}, u_{2} = \frac{- 5 - 57}{2 \cdot 2}

u = \frac{- 5 + 57}{2 \cdot 2} : \frac{- 5 + 57}{4}

\frac{- 5 + 57}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 5 + 57}{4}

u = \frac{- 5 - 57}{2 \cdot 2} : \frac{- 5 - 57}{4}

\frac{- 5 - 57}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 5 - 57}{4}

二次方程组的解是:

u = \frac{- 5 + 57}{4}, u = \frac{- 5 - 57}{4}

u = cos (x)

代回

cos (x) = \frac{- 5 + 57}{4}, cos (x) = \frac{- 5 - 57}{4}

cos (x) = \frac{- 5 + 57}{4}, cos (x) = \frac{- 5 - 57}{4}

cos (x) = \frac{- 5 + 57}{4} : x = arccos (\frac{- 5 + 57}{4}) + 2 πn, x = 2 π - arccos (\frac{- 5 + 57}{4}) + 2 πn

cos (x) = \frac{- 5 + 57}{4}

使用反三角函数性质

cos (x) = \frac{- 5 + 57}{4}

cos (x) = \frac{- 5 + 57}{4}

的通解

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

x = arccos (\frac{- 5 + 57}{4}) + 2 πn, x = 2 π - arccos (\frac{- 5 + 57}{4}) + 2 πn

x = arccos (\frac{- 5 + 57}{4}) + 2 πn, x = 2 π - arccos (\frac{- 5 + 57}{4}) + 2 πn

cos (x) = \frac{- 5 - 57}{4} :

无解

cos (x) = \frac{- 5 - 57}{4}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{- 5 + 57}{4}) + 2 πn, x = 2 π - arccos (\frac{- 5 + 57}{4}) + 2 πn

以小数形式表示解

x = 0.87960 \dots + 2 πn, x = 2 π - 0.87960 \dots + 2 πn

作图

流行的例子

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