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受欢迎的三角函数 >

sqrt(5)cos(x)-2sin(x)=0.5

解答

5 cos (x) - 2 sin (x) = 0.5

解答

x = π + 1.00851 \dots + 2 πn, x = 0.67362 \dots + 2 πn

+1

度数

x = 237.78375 \dots^{\circ} + 36 0^{\circ} n, x = 38.59561 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 cos (x) - 2 sin (x) = 0.5

两边加上

2 sin (x)

5 cos (x) = 0.5 + 2 sin (x)

两边进行平方

(5 cos (x))^{2} = (0.5 + 2 sin (x))^{2}

两边减去

(0.5 + 2 sin (x))^{2}

5 cos^{2} (x) - 0.25 - 2 sin (x) - 4 sin^{2} (x) = 0

使用三角恒等式改写

- 0.25 - 2 sin (x) - 4 sin^{2} (x) + 5 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)

= - 0.25 - 2 sin (x) - 4 sin^{2} (x) + 5 (1 - sin^{2} (x))

化简

- 0.25 - 2 sin (x) - 4 sin^{2} (x) + 5 (1 - sin^{2} (x)) : - 9 sin^{2} (x) - 2 sin (x) + 4.75

- 0.25 - 2 sin (x) - 4 sin^{2} (x) + 5 (1 - sin^{2} (x))

乘开

5 (1 - sin^{2} (x)) : 5 - 5 sin^{2} (x)

5 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 5, b = 1, c = sin^{2} (x)

= 5 \cdot 1 - 5 sin^{2} (x)

数字相乘:

5 \cdot 1 = 5

= 5 - 5 sin^{2} (x)

= - 0.25 - 2 sin (x) - 4 sin^{2} (x) + 5 - 5 sin^{2} (x)

化简

- 0.25 - 2 sin (x) - 4 sin^{2} (x) + 5 - 5 sin^{2} (x) : - 9 sin^{2} (x) - 2 sin (x) + 4.75

- 0.25 - 2 sin (x) - 4 sin^{2} (x) + 5 - 5 sin^{2} (x)

对同类项分组

= - 2 sin (x) - 4 sin^{2} (x) - 5 sin^{2} (x) - 0.25 + 5

同类项相加:

- 4 sin^{2} (x) - 5 sin^{2} (x) = - 9 sin^{2} (x)

= - 2 sin (x) - 9 sin^{2} (x) - 0.25 + 5

数字相加/相减:

- 0.25 + 5 = 4.75

= - 9 sin^{2} (x) - 2 sin (x) + 4.75

= - 9 sin^{2} (x) - 2 sin (x) + 4.75

= - 9 sin^{2} (x) - 2 sin (x) + 4.75

4.75 - 2 sin (x) - 9 sin^{2} (x) = 0

用替代法求解

4.75 - 2 sin (x) - 9 sin^{2} (x) = 0

令

: sin (x) = u

4.75 - 2 u - 9 u^{2} = 0

4.75 - 2 u - 9 u^{2} = 0 : u = - \frac{2 + 5 7}{18}, u = \frac{5 7 - 2}{18}

4.75 - 2 u - 9 u^{2} = 0

在两边乘以

100

4.75 - 2 u - 9 u^{2} = 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

4.75 \cdot 100 - 2 u \cdot 100 - 9 u^{2} \cdot 100 = 0 \cdot 100

整理后得

475 - 200 u - 900 u^{2} = 0

475 - 200 u - 900 u^{2} = 0

改写成标准形式

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

- 900 u^{2} - 200 u + 475 = 0

使用求根公式求解

- 900 u^{2} - 200 u + 475 = 0

二次方程求根公式:

若

a = - 900, b = - 200, c = 475

u_{1, 2} = \frac{- ( - 200 ) \pm ( - 200 ) ^{2} - 4 ( - 900 ) \cdot 475}{2 ( - 900 )}

u_{1, 2} = \frac{- ( - 200 ) \pm ( - 200 ) ^{2} - 4 ( - 900 ) \cdot 475}{2 ( - 900 )}

(- 200)^{2} - 4 (- 900) \cdot 475 = 5007

(- 200)^{2} - 4 (- 900) \cdot 475

使用法则

- (- a) = a

= (- 200)^{2} + 4 \cdot 900 \cdot 475

使用指数法则:

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

若

n

是偶数

(- 200)^{2} = 20 0^{2}

= 20 0^{2} + 4 \cdot 900 \cdot 475

数字相乘:

4 \cdot 900 \cdot 475 = 1710000

= 20 0^{2} + 1710000

20 0^{2} = 40000

= 40000 + 1710000

数字相加:

40000 + 1710000 = 1750000

= 1750000

1750000

质因数分解:

2^{4} \cdot 5^{6} \cdot 7

1750000

= 5^{6} \cdot 2^{4} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{4} 5^{6}

使用根式运算法则:

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

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

= 2^{2} 7 5^{6}

使用根式运算法则:

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

5^{6} = 5^{\frac{6}{2}} = 5^{3}

= 5^{3} \cdot 2^{2} 7

整理后得

= 5007

u_{1, 2} = \frac{- ( - 200 ) \pm 500 7}{2 ( - 900 )}

将解分隔开

u_{1} = \frac{- ( - 200 ) + 500 7}{2 ( - 900 )}, u_{2} = \frac{- ( - 200 ) - 500 7}{2 ( - 900 )}

u = \frac{- ( - 200 ) + 500 7}{2 ( - 900 )} : - \frac{2 + 5 7}{18}

\frac{- ( - 200 ) + 500 7}{2 ( - 900 )}

去除括号:

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

= \frac{200 + 500 7}{- 2 \cdot 900}

数字相乘:

2 \cdot 900 = 1800

= \frac{200 + 500 7}{- 1800}

使用分式法则:

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

= - \frac{200 + 500 7}{1800}

消掉

\frac{200 + 500 7}{1800} : \frac{2 + 5 7}{18}

\frac{200 + 500 7}{1800}

分解

200 + 5007 : 100 (2 + 57)

200 + 5007

改写为

= 100 \cdot 2 + 100 \cdot 57

因式分解出通项

100

= 100 (2 + 57)

= \frac{100 ( 2 + 5 7 )}{1800}

约分:

100

= \frac{2 + 5 7}{18}

= - \frac{2 + 5 7}{18}

u = \frac{- ( - 200 ) - 500 7}{2 ( - 900 )} : \frac{5 7 - 2}{18}

\frac{- ( - 200 ) - 500 7}{2 ( - 900 )}

去除括号:

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

= \frac{200 - 500 7}{- 2 \cdot 900}

数字相乘:

2 \cdot 900 = 1800

= \frac{200 - 500 7}{- 1800}

使用分式法则:

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

200 - 5007 = - (5007 - 200)

= \frac{500 7 - 200}{1800}

分解

5007 - 200 : 100 (57 - 2)

5007 - 200

改写为

= 100 \cdot 57 - 100 \cdot 2

因式分解出通项

100

= 100 (57 - 2)

= \frac{100 ( 5 7 - 2 )}{1800}

约分:

100

= \frac{5 7 - 2}{18}

二次方程组的解是:

u = - \frac{2 + 5 7}{18}, u = \frac{5 7 - 2}{18}

u = sin (x)

代回

sin (x) = - \frac{2 + 5 7}{18}, sin (x) = \frac{5 7 - 2}{18}

sin (x) = - \frac{2 + 5 7}{18}, sin (x) = \frac{5 7 - 2}{18}

sin (x) = - \frac{2 + 5 7}{18} : x = arcsin (- \frac{2 + 5 7}{18}) + 2 πn, x = π + arcsin (\frac{2 + 5 7}{18}) + 2 πn

sin (x) = - \frac{2 + 5 7}{18}

使用反三角函数性质

sin (x) = - \frac{2 + 5 7}{18}

sin (x) = - \frac{2 + 5 7}{18}

的通解

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

x = arcsin (- \frac{2 + 5 7}{18}) + 2 πn, x = π + arcsin (\frac{2 + 5 7}{18}) + 2 πn

x = arcsin (- \frac{2 + 5 7}{18}) + 2 πn, x = π + arcsin (\frac{2 + 5 7}{18}) + 2 πn

sin (x) = \frac{5 7 - 2}{18} : x = arcsin (\frac{5 7 - 2}{18}) + 2 πn, x = π - arcsin (\frac{5 7 - 2}{18}) + 2 πn

sin (x) = \frac{5 7 - 2}{18}

使用反三角函数性质

sin (x) = \frac{5 7 - 2}{18}

sin (x) = \frac{5 7 - 2}{18}

的通解

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

x = arcsin (\frac{5 7 - 2}{18}) + 2 πn, x = π - arcsin (\frac{5 7 - 2}{18}) + 2 πn

x = arcsin (\frac{5 7 - 2}{18}) + 2 πn, x = π - arcsin (\frac{5 7 - 2}{18}) + 2 πn

合并所有解

x = arcsin (- \frac{2 + 5 7}{18}) + 2 πn, x = π + arcsin (\frac{2 + 5 7}{18}) + 2 πn, x = arcsin (\frac{5 7 - 2}{18}) + 2 πn, x = π - arcsin (\frac{5 7 - 2}{18}) + 2 πn

将解代入原方程进行验证

将它们代入

5 cos (x) - 2 sin (x) = 0.5

检验解是否符合

去除与方程不符的解。

检验

arcsin (- \frac{2 + 5 7}{18}) + 2 πn

的解:

假

arcsin (- \frac{2 + 5 7}{18}) + 2 πn

代入

n = 1

arcsin (- \frac{2 + 5 7}{18}) + 2 π 1

对于

5 cos (x) - 2 sin (x) = 0.5

代入

x = arcsin (- \frac{2 + 5 7}{18}) + 2 π 1

5 cos (arcsin (- \frac{2 + 5 7}{18}) + 2 π 1) - 2 sin (arcsin (- \frac{2 + 5 7}{18}) + 2 π 1) = 0.5

整理后得

2.88416 \dots = 0.5

\Rightarrow 假

检验

π + arcsin (\frac{2 + 5 7}{18}) + 2 πn

的解:

真

π + arcsin (\frac{2 + 5 7}{18}) + 2 πn

代入

n = 1

π + arcsin (\frac{2 + 5 7}{18}) + 2 π 1

对于

5 cos (x) - 2 sin (x) = 0.5

代入

x = π + arcsin (\frac{2 + 5 7}{18}) + 2 π 1

5 cos (π + arcsin (\frac{2 + 5 7}{18}) + 2 π 1) - 2 sin (π + arcsin (\frac{2 + 5 7}{18}) + 2 π 1) = 0.5

整理后得

0.5 = 0.5

\Rightarrow 真

检验

arcsin (\frac{5 7 - 2}{18}) + 2 πn

的解:

真

arcsin (\frac{5 7 - 2}{18}) + 2 πn

代入

n = 1

arcsin (\frac{5 7 - 2}{18}) + 2 π 1

对于

5 cos (x) - 2 sin (x) = 0.5

代入

x = arcsin (\frac{5 7 - 2}{18}) + 2 π 1

5 cos (arcsin (\frac{5 7 - 2}{18}) + 2 π 1) - 2 sin (arcsin (\frac{5 7 - 2}{18}) + 2 π 1) = 0.5

整理后得

0.5 = 0.5

\Rightarrow 真

检验

π - arcsin (\frac{5 7 - 2}{18}) + 2 πn

的解:

假

π - arcsin (\frac{5 7 - 2}{18}) + 2 πn

代入

n = 1

π - arcsin (\frac{5 7 - 2}{18}) + 2 π 1

对于

5 cos (x) - 2 sin (x) = 0.5

代入

x = π - arcsin (\frac{5 7 - 2}{18}) + 2 π 1

5 cos (π - arcsin (\frac{5 7 - 2}{18}) + 2 π 1) - 2 sin (π - arcsin (\frac{5 7 - 2}{18}) + 2 π 1) = 0.5

整理后得

- 2.99527 \dots = 0.5

\Rightarrow 假

x = π + arcsin (\frac{2 + 5 7}{18}) + 2 πn, x = arcsin (\frac{5 7 - 2}{18}) + 2 πn

以小数形式表示解

x = π + 1.00851 \dots + 2 πn, x = 0.67362 \dots + 2 πn

作图

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