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受欢迎的三角函数 >

6sin(x)-2cos(x)=7

解答

6 sin (x) - 2 cos (x) = 7

解答

x \in R 无解

求解步骤

6 sin (x) - 2 cos (x) = 7

两边加上

2 cos (x)

6 sin (x) = 7 + 2 cos (x)

两边进行平方

(6 sin (x))^{2} = (7 + 2 cos (x))^{2}

两边减去

(7 + 2 cos (x))^{2}

36 sin^{2} (x) - 49 - 28 cos (x) - 4 cos^{2} (x) = 0

使用三角恒等式改写

- 49 - 28 cos (x) + 36 sin^{2} (x) - 4 cos^{2} (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

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

= - 49 - 28 cos (x) + 36 (1 - cos^{2} (x)) - 4 cos^{2} (x)

化简

- 49 - 28 cos (x) + 36 (1 - cos^{2} (x)) - 4 cos^{2} (x) : - 40 cos^{2} (x) - 28 cos (x) - 13

- 49 - 28 cos (x) + 36 (1 - cos^{2} (x)) - 4 cos^{2} (x)

乘开

36 (1 - cos^{2} (x)) : 36 - 36 cos^{2} (x)

36 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 36, b = 1, c = cos^{2} (x)

= 36 \cdot 1 - 36 cos^{2} (x)

数字相乘:

36 \cdot 1 = 36

= 36 - 36 cos^{2} (x)

= - 49 - 28 cos (x) + 36 - 36 cos^{2} (x) - 4 cos^{2} (x)

化简

- 49 - 28 cos (x) + 36 - 36 cos^{2} (x) - 4 cos^{2} (x) : - 40 cos^{2} (x) - 28 cos (x) - 13

- 49 - 28 cos (x) + 36 - 36 cos^{2} (x) - 4 cos^{2} (x)

同类项相加:

- 36 cos^{2} (x) - 4 cos^{2} (x) = - 40 cos^{2} (x)

= - 49 - 28 cos (x) + 36 - 40 cos^{2} (x)

对同类项分组

= - 28 cos (x) - 40 cos^{2} (x) - 49 + 36

数字相加/相减:

- 49 + 36 = - 13

= - 40 cos^{2} (x) - 28 cos (x) - 13

= - 40 cos^{2} (x) - 28 cos (x) - 13

= - 40 cos^{2} (x) - 28 cos (x) - 13

- 13 - 28 cos (x) - 40 cos^{2} (x) = 0

用替代法求解

- 13 - 28 cos (x) - 40 cos^{2} (x) = 0

令

: cos (x) = u

- 13 - 28 u - 40 u^{2} = 0

- 13 - 28 u - 40 u^{2} = 0 : u = - \frac{7}{20} - i \frac{9}{20}, u = - \frac{7}{20} + i \frac{9}{20}

- 13 - 28 u - 40 u^{2} = 0

改写成标准形式

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

- 40 u^{2} - 28 u - 13 = 0

使用求根公式求解

- 40 u^{2} - 28 u - 13 = 0

二次方程求根公式:

若

a = - 40, b = - 28, c = - 13

u_{1, 2} = \frac{- ( - 28 ) \pm ( - 28 ) ^{2} - 4 ( - 40 ) ( - 13 )}{2 ( - 40 )}

u_{1, 2} = \frac{- ( - 28 ) \pm ( - 28 ) ^{2} - 4 ( - 40 ) ( - 13 )}{2 ( - 40 )}

化简

(- 28)^{2} - 4 (- 40) (- 13) : 36 i

(- 28)^{2} - 4 (- 40) (- 13)

使用法则

- (- a) = a

= (- 28)^{2} - 4 \cdot 40 \cdot 13

使用指数法则:

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

若

n

是偶数

(- 28)^{2} = 2 8^{2}

= 2 8^{2} - 4 \cdot 40 \cdot 13

数字相乘:

4 \cdot 40 \cdot 13 = 2080

= 2 8^{2} - 2080

使用虚数运算法则:

- a = i a

= i 2080 - 2 8^{2}

- 2 8^{2} + 2080 = 36

- 2 8^{2} + 2080

2 8^{2} = 784

= - 784 + 2080

数字相加/相减:

- 784 + 2080 = 1296

= 1296

因式分解数字:

1296 = 3 6^{2}

= 3 6^{2}

使用根式运算法则:

3 6^{2} = 36

= 36

= 36 i

u_{1, 2} = \frac{- ( - 28 ) \pm 36 i}{2 ( - 40 )}

将解分隔开

u_{1} = \frac{- ( - 28 ) + 36 i}{2 ( - 40 )}, u_{2} = \frac{- ( - 28 ) - 36 i}{2 ( - 40 )}

u = \frac{- ( - 28 ) + 36 i}{2 ( - 40 )} : - \frac{7}{20} - i \frac{9}{20}

\frac{- ( - 28 ) + 36 i}{2 ( - 40 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{28 + 36 i}{- 2 \cdot 40}

数字相乘:

2 \cdot 40 = 80

= \frac{28 + 36 i}{- 80}

使用分式法则:

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

= - \frac{28 + 36 i}{80}

消掉

\frac{28 + 36 i}{80} : \frac{7 + 9 i}{20}

\frac{28 + 36 i}{80}

分解

28 + 36 i : 4 (7 + 9 i)

28 + 36 i

改写为

= 4 \cdot 7 + 4 \cdot 9 i

因式分解出通项

4

= 4 (7 + 9 i)

= \frac{4 ( 7 + 9 i )}{80}

约分:

4

= \frac{7 + 9 i}{20}

= - \frac{7 + 9 i}{20}

将

- \frac{7 + 9 i}{20}

改写成标准复数形式:

- \frac{7}{20} - \frac{9}{20} i

- \frac{7 + 9 i}{20}

使用分式法则:

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

\frac{7 + 9 i}{20} = - (\frac{7}{20}) - (\frac{9 i}{20})

= - (\frac{7}{20}) - (\frac{9 i}{20})

去除括号:

(a) = a

= - \frac{7}{20} - \frac{9 i}{20}

= - \frac{7}{20} - \frac{9}{20} i

u = \frac{- ( - 28 ) - 36 i}{2 ( - 40 )} : - \frac{7}{20} + i \frac{9}{20}

\frac{- ( - 28 ) - 36 i}{2 ( - 40 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{28 - 36 i}{- 2 \cdot 40}

数字相乘:

2 \cdot 40 = 80

= \frac{28 - 36 i}{- 80}

使用分式法则:

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

= - \frac{28 - 36 i}{80}

消掉

\frac{28 - 36 i}{80} : \frac{7 - 9 i}{20}

\frac{28 - 36 i}{80}

分解

28 - 36 i : 4 (7 - 9 i)

28 - 36 i

改写为

= 4 \cdot 7 - 4 \cdot 9 i

因式分解出通项

4

= 4 (7 - 9 i)

= \frac{4 ( 7 - 9 i )}{80}

约分:

4

= \frac{7 - 9 i}{20}

= - \frac{7 - 9 i}{20}

将

- \frac{7 - 9 i}{20}

改写成标准复数形式:

- \frac{7}{20} + \frac{9}{20} i

- \frac{7 - 9 i}{20}

使用分式法则:

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

\frac{7 - 9 i}{20} = - (\frac{7}{20}) - (- \frac{9 i}{20})

= - (\frac{7}{20}) - (- \frac{9 i}{20})

去除括号:

(a) = a, - (- a) = a

= - \frac{7}{20} + \frac{9 i}{20}

= - \frac{7}{20} + \frac{9}{20} i

二次方程组的解是:

u = - \frac{7}{20} - i \frac{9}{20}, u = - \frac{7}{20} + i \frac{9}{20}

u = cos (x)

代回

cos (x) = - \frac{7}{20} - i \frac{9}{20}, cos (x) = - \frac{7}{20} + i \frac{9}{20}

cos (x) = - \frac{7}{20} - i \frac{9}{20}, cos (x) = - \frac{7}{20} + i \frac{9}{20}

cos (x) = - \frac{7}{20} - i \frac{9}{20} :

无解

cos (x) = - \frac{7}{20} - i \frac{9}{20}

无解

cos (x) = - \frac{7}{20} + i \frac{9}{20} :

无解

cos (x) = - \frac{7}{20} + i \frac{9}{20}

无解

合并所有解

无解

将解代入原方程进行验证

将它们代入

6 sin (x) - 2 cos (x) = 7

检验解是否符合

去除与方程不符的解。

x \in R 无解

作图

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