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受欢迎的三角函数 >

6193cos(x)+2880cos(2x)=0

解答

6193 cos (x) + 2880 cos (2 x) = 0

解答

x = 1.21251 \dots + 2 πn, x = 2 π - 1.21251 \dots + 2 πn

+1

度数

x = 69.47171 \dots^{\circ} + 36 0^{\circ} n, x = 290.52828 \dots^{\circ} + 36 0^{\circ} n

求解步骤

6193 cos (x) + 2880 cos (2 x) = 0

使用三角恒等式改写

2880 cos (2 x) + 6193 cos (x)

使用倍角公式:

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

= 2880 (2 cos^{2} (x) - 1) + 6193 cos (x)

(- 1 + 2 cos^{2} (x)) \cdot 2880 + 6193 cos (x) = 0

用替代法求解

(- 1 + 2 cos^{2} (x)) \cdot 2880 + 6193 cos (x) = 0

令

: cos (x) = u

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u = 0

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u = 0 : u = \frac{- 6193 + 104708449}{11520}, u = \frac{- 6193 - 104708449}{11520}

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u = 0

展开

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u : - 2880 + 5760 u^{2} + 6193 u

(- 1 + 2 u^{2}) \cdot 2880 + 6193 u

= 2880 (- 1 + 2 u^{2}) + 6193 u

乘开

2880 (- 1 + 2 u^{2}) : - 2880 + 5760 u^{2}

2880 (- 1 + 2 u^{2})

使用分配律:

a (b + c) = ab + a c

a = 2880, b = - 1, c = 2 u^{2}

= 2880 (- 1) + 2880 \cdot 2 u^{2}

使用加减运算法则

+ (- a) = - a

= - 2880 \cdot 1 + 2880 \cdot 2 u^{2}

化简

- 2880 \cdot 1 + 2880 \cdot 2 u^{2} : - 2880 + 5760 u^{2}

- 2880 \cdot 1 + 2880 \cdot 2 u^{2}

数字相乘:

2880 \cdot 1 = 2880

= - 2880 + 2880 \cdot 2 u^{2}

数字相乘:

2880 \cdot 2 = 5760

= - 2880 + 5760 u^{2}

= - 2880 + 5760 u^{2}

= - 2880 + 5760 u^{2} + 6193 u

- 2880 + 5760 u^{2} + 6193 u = 0

两边除以

5760

- \frac{2880}{5760} + \frac{5760 u ^{2}}{5760} + \frac{6193 u}{5760} = \frac{0}{5760}

改写成标准形式

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

u^{2} + \frac{6193 u}{5760} - \frac{1}{2} = 0

使用求根公式求解

u^{2} + \frac{6193 u}{5760} - \frac{1}{2} = 0

二次方程求根公式:

若

a = 1, b = \frac{6193}{5760}, c = - \frac{1}{2}

u_{1, 2} = \frac{- \frac{6193}{5760} \pm ( \frac{6193}{5760} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{1}{2} )}{2 \cdot 1}

u_{1, 2} = \frac{- \frac{6193}{5760} \pm ( \frac{6193}{5760} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{1}{2} )}{2 \cdot 1}

(\frac{6193}{5760})^{2} - 4 \cdot 1 \cdot (- \frac{1}{2}) = \frac{104708449}{5760}

(\frac{6193}{5760})^{2} - 4 \cdot 1 \cdot (- \frac{1}{2})

使用法则

- (- a) = a

= (\frac{6193}{5760})^{2} + 4 \cdot 1 \cdot \frac{1}{2}

(\frac{6193}{5760})^{2} = \frac{619 3 ^{2}}{576 0 ^{2}}

(\frac{6193}{5760})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{619 3 ^{2}}{576 0 ^{2}}

4 \cdot 1 \cdot \frac{1}{2} = 2

4 \cdot 1 \cdot \frac{1}{2}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 1 \cdot \frac{1 \cdot 4}{2}

\frac{1 \cdot 4}{2} = 2

\frac{1 \cdot 4}{2}

数字相乘:

1 \cdot 4 = 4

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

= 1 \cdot 2

数字相乘:

1 \cdot 2 = 2

= 2

= \frac{619 3 ^{2}}{576 0 ^{2}} + 2

\frac{619 3 ^{2}}{576 0 ^{2}} = \frac{38353249}{33177600}

\frac{619 3 ^{2}}{576 0 ^{2}}

619 3^{2} = 38353249

= \frac{38353249}{576 0 ^{2}}

576 0^{2} = 33177600

= \frac{38353249}{33177600}

= \frac{38353249}{33177600} + 2

化简

\frac{38353249}{33177600} + 2 : \frac{104708449}{33177600}

\frac{38353249}{33177600} + 2

将项转换为分式:

2 = \frac{2 \cdot 33177600}{33177600}

= \frac{2 \cdot 33177600}{33177600} + \frac{38353249}{33177600}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{2 \cdot 33177600 + 38353249}{33177600}

2 \cdot 33177600 + 38353249 = 104708449

2 \cdot 33177600 + 38353249

数字相乘:

2 \cdot 33177600 = 66355200

= 66355200 + 38353249

数字相加:

66355200 + 38353249 = 104708449

= 104708449

= \frac{104708449}{33177600}

= \frac{104708449}{33177600}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{104708449}{33177600}

33177600 = 5760

33177600

因式分解数字:

33177600 = 576 0^{2}

= 576 0^{2}

使用根式运算法则:

n a^{n} = a

576 0^{2} = 5760

= 5760

= \frac{104708449}{5760}

u_{1, 2} = \frac{- \frac{6193}{5760} \pm \frac{104708449}{5760}}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- \frac{6193}{5760} + \frac{104708449}{5760}}{2 \cdot 1}, u_{2} = \frac{- \frac{6193}{5760} - \frac{104708449}{5760}}{2 \cdot 1}

u = \frac{- \frac{6193}{5760} + \frac{104708449}{5760}}{2 \cdot 1} : \frac{- 6193 + 104708449}{11520}

\frac{- \frac{6193}{5760} + \frac{104708449}{5760}}{2 \cdot 1}

合并分式

- \frac{6193}{5760} + \frac{104708449}{5760} : \frac{- 6193 + 104708449}{5760}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 6193 + 104708449}{5760}

= \frac{\frac{- 6193 + 104708449}{5760}}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{\frac{- 6193 + 104708449}{5760}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- 6193 + 104708449}{5760 \cdot 2}

数字相乘:

5760 \cdot 2 = 11520

= \frac{- 6193 + 104708449}{11520}

u = \frac{- \frac{6193}{5760} - \frac{104708449}{5760}}{2 \cdot 1} : \frac{- 6193 - 104708449}{11520}

\frac{- \frac{6193}{5760} - \frac{104708449}{5760}}{2 \cdot 1}

合并分式

- \frac{6193}{5760} - \frac{104708449}{5760} : \frac{- 6193 - 104708449}{5760}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 6193 - 104708449}{5760}

= \frac{\frac{- 6193 - 104708449}{5760}}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{\frac{- 6193 - 104708449}{5760}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- 6193 - 104708449}{5760 \cdot 2}

数字相乘:

5760 \cdot 2 = 11520

= \frac{- 6193 - 104708449}{11520}

二次方程组的解是:

u = \frac{- 6193 + 104708449}{11520}, u = \frac{- 6193 - 104708449}{11520}

u = cos (x)

代回

cos (x) = \frac{- 6193 + 104708449}{11520}, cos (x) = \frac{- 6193 - 104708449}{11520}

cos (x) = \frac{- 6193 + 104708449}{11520}, cos (x) = \frac{- 6193 - 104708449}{11520}

cos (x) = \frac{- 6193 + 104708449}{11520} : x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

cos (x) = \frac{- 6193 + 104708449}{11520}

使用反三角函数性质

cos (x) = \frac{- 6193 + 104708449}{11520}

cos (x) = \frac{- 6193 + 104708449}{11520}

的通解

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

x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

cos (x) = \frac{- 6193 - 104708449}{11520} :

无解

cos (x) = \frac{- 6193 - 104708449}{11520}

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn, x = 2 π - arccos (\frac{- 6193 + 104708449}{11520}) + 2 πn

以小数形式表示解

x = 1.21251 \dots + 2 πn, x = 2 π - 1.21251 \dots + 2 πn

作图

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