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受欢迎的三角函数 >

0=1.5961cos(x)+0.0422cos(2*x)

解答

0 = 1.5961 \cdot cos (x) + 0.0422 \cdot cos (2 \cdot x)

解答

x = 1.54439 \dots + 2 πn, x = 2 π - 1.54439 \dots + 2 πn

+1

度数

x = 88.48706 \dots^{\circ} + 36 0^{\circ} n, x = 271.51293 \dots^{\circ} + 36 0^{\circ} n

求解步骤

0 = 1.5961 cos (x) + 0.0422 cos (2 x)

交换两边

1.5961 cos (x) + 0.0422 cos (2 x) = 0

使用三角恒等式改写

0.0422 cos (2 x) + 1.5961 cos (x)

使用倍角公式:

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

= 0.0422 (2 cos^{2} (x) - 1) + 1.5961 cos (x)

(- 1 + 2 cos^{2} (x)) \cdot 0.0422 + 1.5961 cos (x) = 0

用替代法求解

(- 1 + 2 cos^{2} (x)) \cdot 0.0422 + 1.5961 cos (x) = 0

令

: cos (x) = u

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.5961 u = 0

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.5961 u = 0 : u = \frac{- 1.5961 + 2.56178193}{0.1688}, u = \frac{- 1.5961 - 2.56178193}{0.1688}

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.5961 u = 0

展开

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.5961 u : - 0.0422 + 0.0844 u^{2} + 1.5961 u

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.5961 u

= 0.0422 (- 1 + 2 u^{2}) + 1.5961 u

乘开

0.0422 (- 1 + 2 u^{2}) : - 0.0422 + 0.0844 u^{2}

0.0422 (- 1 + 2 u^{2})

使用分配律:

a (b + c) = ab + a c

a = 0.0422, b = - 1, c = 2 u^{2}

= 0.0422 (- 1) + 0.0422 \cdot 2 u^{2}

使用加减运算法则

+ (- a) = - a

= - 1 \cdot 0.0422 + 2 \cdot 0.0422 u^{2}

化简

- 1 \cdot 0.0422 + 2 \cdot 0.0422 u^{2} : - 0.0422 + 0.0844 u^{2}

- 1 \cdot 0.0422 + 2 \cdot 0.0422 u^{2}

数字相乘:

1 \cdot 0.0422 = 0.0422

= - 0.0422 + 2 \cdot 0.0422 u^{2}

数字相乘:

2 \cdot 0.0422 = 0.0844

= - 0.0422 + 0.0844 u^{2}

= - 0.0422 + 0.0844 u^{2}

= - 0.0422 + 0.0844 u^{2} + 1.5961 u

- 0.0422 + 0.0844 u^{2} + 1.5961 u = 0

改写成标准形式

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

0.0844 u^{2} + 1.5961 u - 0.0422 = 0

使用求根公式求解

0.0844 u^{2} + 1.5961 u - 0.0422 = 0

二次方程求根公式:

若

a = 0.0844, b = 1.5961, c = - 0.0422

u_{1, 2} = \frac{- 1.5961 \pm 1.596 1 ^{2} - 4 \cdot 0.0844 ( - 0.0422 )}{2 \cdot 0.0844}

u_{1, 2} = \frac{- 1.5961 \pm 1.596 1 ^{2} - 4 \cdot 0.0844 ( - 0.0422 )}{2 \cdot 0.0844}

1.596 1^{2} - 4 \cdot 0.0844 (- 0.0422) = 2.56178193

1.596 1^{2} - 4 \cdot 0.0844 (- 0.0422)

使用法则

- (- a) = a

= 1.596 1^{2} + 4 \cdot 0.0844 \cdot 0.0422

数字相乘:

4 \cdot 0.0844 \cdot 0.0422 = 0.01424672

= 1.596 1^{2} + 0.01424672

1.596 1^{2} = 2.54753521

= 2.54753521 + 0.01424672

数字相加:

2.54753521 + 0.01424672 = 2.56178193

= 2.56178193

u_{1, 2} = \frac{- 1.5961 \pm 2.56178193}{2 \cdot 0.0844}

将解分隔开

u_{1} = \frac{- 1.5961 + 2.56178193}{2 \cdot 0.0844}, u_{2} = \frac{- 1.5961 - 2.56178193}{2 \cdot 0.0844}

u = \frac{- 1.5961 + 2.56178193}{2 \cdot 0.0844} : \frac{- 1.5961 + 2.56178193}{0.1688}

\frac{- 1.5961 + 2.56178193}{2 \cdot 0.0844}

数字相乘:

2 \cdot 0.0844 = 0.1688

= \frac{- 1.5961 + 2.56178193}{0.1688}

u = \frac{- 1.5961 - 2.56178193}{2 \cdot 0.0844} : \frac{- 1.5961 - 2.56178193}{0.1688}

\frac{- 1.5961 - 2.56178193}{2 \cdot 0.0844}

数字相乘:

2 \cdot 0.0844 = 0.1688

= \frac{- 1.5961 - 2.56178193}{0.1688}

二次方程组的解是:

u = \frac{- 1.5961 + 2.56178193}{0.1688}, u = \frac{- 1.5961 - 2.56178193}{0.1688}

u = cos (x)

代回

cos (x) = \frac{- 1.5961 + 2.56178193}{0.1688}, cos (x) = \frac{- 1.5961 - 2.56178193}{0.1688}

cos (x) = \frac{- 1.5961 + 2.56178193}{0.1688}, cos (x) = \frac{- 1.5961 - 2.56178193}{0.1688}

cos (x) = \frac{- 1.5961 + 2.56178193}{0.1688} : x = arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn

cos (x) = \frac{- 1.5961 + 2.56178193}{0.1688}

使用反三角函数性质

cos (x) = \frac{- 1.5961 + 2.56178193}{0.1688}

cos (x) = \frac{- 1.5961 + 2.56178193}{0.1688}

的通解

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

x = arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn

x = arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn

cos (x) = \frac{- 1.5961 - 2.56178193}{0.1688} :

无解

cos (x) = \frac{- 1.5961 - 2.56178193}{0.1688}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.5961 + 2.56178193}{0.1688}) + 2 πn

以小数形式表示解

x = 1.54439 \dots + 2 πn, x = 2 π - 1.54439 \dots + 2 πn

作图

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