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受欢迎的三角函数 >

10=-12sin(x)+1.8cos(x)

解答

10 = - 12 sin (x) + 1.8 cos (x)

解答

x = π + 1.11752 \dots + 2 πn, x = - 0.81974 \dots + 2 πn

+1

度数

x = 244.02949 \dots^{\circ} + 36 0^{\circ} n, x = - 46.96796 \dots^{\circ} + 36 0^{\circ} n

求解步骤

10 = - 12 sin (x) + 1.8 cos (x)

两边加上

12 sin (x)

1.8 cos (x) = 10 + 12 sin (x)

两边进行平方

(1.8 cos (x))^{2} = (10 + 12 sin (x))^{2}

两边减去

(10 + 12 sin (x))^{2}

3.24 cos^{2} (x) - 100 - 240 sin (x) - 144 sin^{2} (x) = 0

使用三角恒等式改写

- 100 - 144 sin^{2} (x) - 240 sin (x) + 3.24 cos^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 100 - 144 sin^{2} (x) - 240 sin (x) + 3.24 (1 - sin^{2} (x))

化简

- 100 - 144 sin^{2} (x) - 240 sin (x) + 3.24 (1 - sin^{2} (x)) : - 147.24 sin^{2} (x) - 240 sin (x) - 96.76

- 100 - 144 sin^{2} (x) - 240 sin (x) + 3.24 (1 - sin^{2} (x))

乘开

3.24 (1 - sin^{2} (x)) : 3.24 - 3.24 sin^{2} (x)

3.24 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 3.24, b = 1, c = sin^{2} (x)

= 3.24 \cdot 1 - 3.24 sin^{2} (x)

= 1 \cdot 3.24 - 3.24 sin^{2} (x)

数字相乘:

1 \cdot 3.24 = 3.24

= 3.24 - 3.24 sin^{2} (x)

= - 100 - 144 sin^{2} (x) - 240 sin (x) + 3.24 - 3.24 sin^{2} (x)

化简

- 100 - 144 sin^{2} (x) - 240 sin (x) + 3.24 - 3.24 sin^{2} (x) : - 147.24 sin^{2} (x) - 240 sin (x) - 96.76

- 100 - 144 sin^{2} (x) - 240 sin (x) + 3.24 - 3.24 sin^{2} (x)

对同类项分组

= - 144 sin^{2} (x) - 240 sin (x) - 3.24 sin^{2} (x) - 100 + 3.24

同类项相加:

- 144 sin^{2} (x) - 3.24 sin^{2} (x) = - 147.24 sin^{2} (x)

= - 147.24 sin^{2} (x) - 240 sin (x) - 100 + 3.24

数字相加/相减:

- 100 + 3.24 = - 96.76

= - 147.24 sin^{2} (x) - 240 sin (x) - 96.76

= - 147.24 sin^{2} (x) - 240 sin (x) - 96.76

= - 147.24 sin^{2} (x) - 240 sin (x) - 96.76

- 96.76 - 147.24 sin^{2} (x) - 240 sin (x) = 0

用替代法求解

- 96.76 - 147.24 sin^{2} (x) - 240 sin (x) = 0

令

: sin (x) = u

- 96.76 - 147.24 u^{2} - 240 u = 0

- 96.76 - 147.24 u^{2} - 240 u = 0 : u = - \frac{1000 + 3 1181}{1227}, u = - \frac{1000 - 3 1181}{1227}

- 96.76 - 147.24 u^{2} - 240 u = 0

在两边乘以

100

- 96.76 - 147.24 u^{2} - 240 u = 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 96.76 \cdot 100 - 147.24 u^{2} \cdot 100 - 240 u \cdot 100 = 0 \cdot 100

整理后得

- 9676 - 14724 u^{2} - 24000 u = 0

- 9676 - 14724 u^{2} - 24000 u = 0

改写成标准形式

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

- 14724 u^{2} - 24000 u - 9676 = 0

使用求根公式求解

- 14724 u^{2} - 24000 u - 9676 = 0

二次方程求根公式:

若

a = - 14724, b = - 24000, c = - 9676

u_{1, 2} = \frac{- ( - 24000 ) \pm ( - 24000 ) ^{2} - 4 ( - 14724 ) ( - 9676 )}{2 ( - 14724 )}

u_{1, 2} = \frac{- ( - 24000 ) \pm ( - 24000 ) ^{2} - 4 ( - 14724 ) ( - 9676 )}{2 ( - 14724 )}

(- 24000)^{2} - 4 (- 14724) (- 9676) = 721181

(- 24000)^{2} - 4 (- 14724) (- 9676)

使用法则

- (- a) = a

= (- 24000)^{2} - 4 \cdot 14724 \cdot 9676

使用指数法则:

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

若

n

是偶数

(- 24000)^{2} = 2400 0^{2}

= 2400 0^{2} - 4 \cdot 14724 \cdot 9676

数字相乘:

4 \cdot 14724 \cdot 9676 = 569877696

= 2400 0^{2} - 569877696

2400 0^{2} = 576000000

= 576000000 - 569877696

数字相减:

576000000 - 569877696 = 6122304

= 6122304

6122304

质因数分解:

2^{6} \cdot 3^{4} \cdot 1181

6122304

= 2^{6} \cdot 3^{4} \cdot 1181

使用根式运算法则:

n ab = n a n b

= 1181 2^{6} 3^{4}

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}}

2^{6} = 2^{\frac{6}{2}} = 2^{3}

= 2^{3} 1181 3^{4}

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}}

3^{4} = 3^{\frac{4}{2}} = 3^{2}

= 2^{3} \cdot 3^{2} 1181

整理后得

= 721181

u_{1, 2} = \frac{- ( - 24000 ) \pm 72 1181}{2 ( - 14724 )}

将解分隔开

u_{1} = \frac{- ( - 24000 ) + 72 1181}{2 ( - 14724 )}, u_{2} = \frac{- ( - 24000 ) - 72 1181}{2 ( - 14724 )}

u = \frac{- ( - 24000 ) + 72 1181}{2 ( - 14724 )} : - \frac{1000 + 3 1181}{1227}

\frac{- ( - 24000 ) + 72 1181}{2 ( - 14724 )}

去除括号:

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

= \frac{24000 + 72 1181}{- 2 \cdot 14724}

数字相乘:

2 \cdot 14724 = 29448

= \frac{24000 + 72 1181}{- 29448}

使用分式法则:

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

= - \frac{24000 + 72 1181}{29448}

消掉

\frac{24000 + 72 1181}{29448} : \frac{1000 + 3 1181}{1227}

\frac{24000 + 72 1181}{29448}

分解

24000 + 721181 : 24 (1000 + 31181)

24000 + 721181

改写为

= 24 \cdot 1000 + 24 \cdot 31181

因式分解出通项

24

= 24 (1000 + 31181)

= \frac{24 ( 1000 + 3 1181 )}{29448}

约分:

24

= \frac{1000 + 3 1181}{1227}

= - \frac{1000 + 3 1181}{1227}

u = \frac{- ( - 24000 ) - 72 1181}{2 ( - 14724 )} : - \frac{1000 - 3 1181}{1227}

\frac{- ( - 24000 ) - 72 1181}{2 ( - 14724 )}

去除括号:

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

= \frac{24000 - 72 1181}{- 2 \cdot 14724}

数字相乘:

2 \cdot 14724 = 29448

= \frac{24000 - 72 1181}{- 29448}

使用分式法则:

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

= - \frac{24000 - 72 1181}{29448}

消掉

\frac{24000 - 72 1181}{29448} : \frac{1000 - 3 1181}{1227}

\frac{24000 - 72 1181}{29448}

分解

24000 - 721181 : 24 (1000 - 31181)

24000 - 721181

改写为

= 24 \cdot 1000 - 24 \cdot 31181

因式分解出通项

24

= 24 (1000 - 31181)

= \frac{24 ( 1000 - 3 1181 )}{29448}

约分:

24

= \frac{1000 - 3 1181}{1227}

= - \frac{1000 - 3 1181}{1227}

二次方程组的解是:

u = - \frac{1000 + 3 1181}{1227}, u = - \frac{1000 - 3 1181}{1227}

u = sin (x)

代回

sin (x) = - \frac{1000 + 3 1181}{1227}, sin (x) = - \frac{1000 - 3 1181}{1227}

sin (x) = - \frac{1000 + 3 1181}{1227}, sin (x) = - \frac{1000 - 3 1181}{1227}

sin (x) = - \frac{1000 + 3 1181}{1227} : x = arcsin (- \frac{1000 + 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 πn

sin (x) = - \frac{1000 + 3 1181}{1227}

使用反三角函数性质

sin (x) = - \frac{1000 + 3 1181}{1227}

sin (x) = - \frac{1000 + 3 1181}{1227}

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{1000 + 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 πn

x = arcsin (- \frac{1000 + 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 πn

sin (x) = - \frac{1000 - 3 1181}{1227} : x = arcsin (- \frac{1000 - 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 πn

sin (x) = - \frac{1000 - 3 1181}{1227}

使用反三角函数性质

sin (x) = - \frac{1000 - 3 1181}{1227}

sin (x) = - \frac{1000 - 3 1181}{1227}

的通解

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{1000 - 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 πn

x = arcsin (- \frac{1000 - 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 πn

合并所有解

x = arcsin (- \frac{1000 + 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 πn, x = arcsin (- \frac{1000 - 3 1181}{1227}) + 2 πn, x = π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 πn

将解代入原方程进行验证

将它们代入

- 12 sin (x) + 1.8 cos (x) = 10

检验解是否符合

去除与方程不符的解。

检验

arcsin (- \frac{1000 + 3 1181}{1227}) + 2 πn

的解:

假

arcsin (- \frac{1000 + 3 1181}{1227}) + 2 πn

代入

n = 1

arcsin (- \frac{1000 + 3 1181}{1227}) + 2 π 1

对于

- 12 sin (x) + 1.8 cos (x) = 10

代入

x = arcsin (- \frac{1000 + 3 1181}{1227}) + 2 π 1

- 12 sin (arcsin (- \frac{1000 + 3 1181}{1227}) + 2 π 1) + 1.8 cos (arcsin (- \frac{1000 + 3 1181}{1227}) + 2 π 1) = 10

整理后得

11.57647 \dots = 10

\Rightarrow 假

检验

π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 πn

的解:

真

π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 πn

代入

n = 1

π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 π 1

对于

- 12 sin (x) + 1.8 cos (x) = 10

代入

x = π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 π 1

- 12 sin (π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 π 1) + 1.8 cos (π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 π 1) = 10

整理后得

10 = 10

\Rightarrow 真

检验

arcsin (- \frac{1000 - 3 1181}{1227}) + 2 πn

的解:

真

arcsin (- \frac{1000 - 3 1181}{1227}) + 2 πn

代入

n = 1

arcsin (- \frac{1000 - 3 1181}{1227}) + 2 π 1

对于

- 12 sin (x) + 1.8 cos (x) = 10

代入

x = arcsin (- \frac{1000 - 3 1181}{1227}) + 2 π 1

- 12 sin (arcsin (- \frac{1000 - 3 1181}{1227}) + 2 π 1) + 1.8 cos (arcsin (- \frac{1000 - 3 1181}{1227}) + 2 π 1) = 10

整理后得

10 = 10

\Rightarrow 真

检验

π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 πn

的解:

假

π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 πn

代入

n = 1

π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 π 1

对于

- 12 sin (x) + 1.8 cos (x) = 10

代入

x = π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 π 1

- 12 sin (π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 π 1) + 1.8 cos (π + arcsin (\frac{1000 - 3 1181}{1227}) + 2 π 1) = 10

整理后得

7.54333 \dots = 10

\Rightarrow 假

x = π + arcsin (\frac{1000 + 3 1181}{1227}) + 2 πn, x = arcsin (- \frac{1000 - 3 1181}{1227}) + 2 πn

以小数形式表示解

x = π + 1.11752 \dots + 2 πn, x = - 0.81974 \dots + 2 πn

作图

流行的例子

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