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受欢迎的三角函数 >

-50cos(x)-86.6025400000000…sin(x)=-50

解答

- 50 cos (x) - 86.6025400000000 \dots sin (x) = - 50

解答

x = 2 πn, x = π - 1.04719 \dots + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 119.99999 \dots^{\circ} + 36 0^{\circ} n

求解步骤

- 50 cos (x) - 86.60254 \dots sin (x) = - 50

两边加上

86.60254 \dots sin (x)

- 50 cos (x) = - 50 + 86.60254 \dots sin (x)

两边进行平方

(- 50 cos (x))^{2} = (- 50 + 86.60254 \dots sin (x))^{2}

两边减去

(- 50 + 86.60254 \dots sin (x))^{2}

2500 cos^{2} (x) - 2500 + 8660.254 sin (x) - 7499.99993 \dots sin^{2} (x) = 0

使用三角恒等式改写

- 2500 + 2500 cos^{2} (x) - 7499.99993 \dots sin^{2} (x) + 8660.254 sin (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)

= - 2500 + 2500 (1 - sin^{2} (x)) - 7499.99993 \dots sin^{2} (x) + 8660.254 sin (x)

化简

- 2500 + 2500 (1 - sin^{2} (x)) - 7499.99993 \dots sin^{2} (x) + 8660.254 sin (x) : 8660.254 sin (x) - 9999.99993 \dots sin^{2} (x)

- 2500 + 2500 (1 - sin^{2} (x)) - 7499.99993 \dots sin^{2} (x) + 8660.254 sin (x)

乘开

2500 (1 - sin^{2} (x)) : 2500 - 2500 sin^{2} (x)

2500 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 2500, b = 1, c = sin^{2} (x)

= 2500 \cdot 1 - 2500 sin^{2} (x)

数字相乘:

2500 \cdot 1 = 2500

= 2500 - 2500 sin^{2} (x)

= - 2500 + 2500 - 2500 sin^{2} (x) - 7499.99993 \dots sin^{2} (x) + 8660.254 sin (x)

化简

- 2500 + 2500 - 2500 sin^{2} (x) - 7499.99993 \dots sin^{2} (x) + 8660.254 sin (x) : 8660.254 sin (x) - 9999.99993 \dots sin^{2} (x)

- 2500 + 2500 - 2500 sin^{2} (x) - 7499.99993 \dots sin^{2} (x) + 8660.254 sin (x)

同类项相加:

- 2500 sin^{2} (x) - 7499.99993 \dots sin^{2} (x) = - 9999.99993 \dots sin^{2} (x)

= - 2500 + 2500 - 9999.99993 \dots sin^{2} (x) + 8660.254 sin (x)

- 2500 + 2500 = 0

= 8660.254 sin (x) - 9999.99993 \dots sin^{2} (x)

= 8660.254 sin (x) - 9999.99993 \dots sin^{2} (x)

= 8660.254 sin (x) - 9999.99993 \dots sin^{2} (x)

8660.254 sin (x) - 9999.99993 \dots sin^{2} (x) = 0

用替代法求解

8660.254 sin (x) - 9999.99993 \dots sin^{2} (x) = 0

令

: sin (x) = u

8660.254 u - 9999.99993 \dots u^{2} = 0

8660.254 u - 9999.99993 \dots u^{2} = 0 : u = 0, u = \frac{17320.508}{19999.99986 \dots}

8660.254 u - 9999.99993 \dots u^{2} = 0

改写成标准形式

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

- 9999.99993 \dots u^{2} + 8660.254 u = 0

使用求根公式求解

- 9999.99993 \dots u^{2} + 8660.254 u = 0

二次方程求根公式:

若

a = - 9999.99993 \dots, b = 8660.254, c = 0

u_{1, 2} = \frac{- 8660.254 \pm 8660.25 4 ^{2} - 4 ( - 9999.99993 \dots ) \cdot 0}{2 ( - 9999.99993 \dots )}

u_{1, 2} = \frac{- 8660.254 \pm 8660.25 4 ^{2} - 4 ( - 9999.99993 \dots ) \cdot 0}{2 ( - 9999.99993 \dots )}

8660.25 4^{2} - 4 (- 9999.99993 \dots) \cdot 0 = 8660.254

8660.25 4^{2} - 4 (- 9999.99993 \dots) \cdot 0

使用法则

- (- a) = a

= 8660.25 4^{2} + 4 \cdot 9999.99993 \dots \cdot 0

使用法则

0 \cdot a = 0

= 8660.25 4^{2} + 0

8660.25 4^{2} + 0 = 8660.25 4^{2}

= 8660.25 4^{2}

使用根式运算法则:

n a^{n} = a,

假定

a \geq 0

= 8660.254

u_{1, 2} = \frac{- 8660.254 \pm 8660.254}{2 ( - 9999.99993 \dots )}

将解分隔开

u_{1} = \frac{- 8660.254 + 8660.254}{2 ( - 9999.99993 \dots )}, u_{2} = \frac{- 8660.254 - 8660.254}{2 ( - 9999.99993 \dots )}

u = \frac{- 8660.254 + 8660.254}{2 ( - 9999.99993 \dots )} : 0

\frac{- 8660.254 + 8660.254}{2 ( - 9999.99993 \dots )}

去除括号:

(- a) = - a

= \frac{- 8660.254 + 8660.254}{- 2 \cdot 9999.99993 \dots}

数字相加/相减:

- 8660.254 + 8660.254 = 0

= \frac{0}{- 2 \cdot 9999.99993 \dots}

数字相乘:

2 \cdot 9999.99993 \dots = 19999.99986 \dots

= \frac{0}{- 19999.99986 \dots}

使用分式法则:

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

= - \frac{0}{19999.99986 \dots}

使用法则

\frac{0}{a} = 0, a \neq = 0

= - 0

= 0

u = \frac{- 8660.254 - 8660.254}{2 ( - 9999.99993 \dots )} : \frac{17320.508}{19999.99986 \dots}

\frac{- 8660.254 - 8660.254}{2 ( - 9999.99993 \dots )}

去除括号:

(- a) = - a

= \frac{- 8660.254 - 8660.254}{- 2 \cdot 9999.99993 \dots}

数字相减:

- 8660.254 - 8660.254 = - 17320.508

= \frac{- 17320.508}{- 2 \cdot 9999.99993 \dots}

数字相乘:

2 \cdot 9999.99993 \dots = 19999.99986 \dots

= \frac{- 17320.508}{- 19999.99986 \dots}

使用分式法则:

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

= \frac{17320.508}{19999.99986 \dots}

二次方程组的解是:

u = 0, u = \frac{17320.508}{19999.99986 \dots}

u = sin (x)

代回

sin (x) = 0, sin (x) = \frac{17320.508}{19999.99986 \dots}

sin (x) = 0, sin (x) = \frac{17320.508}{19999.99986 \dots}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

的通解

sin (x)

周期表（周期为

2 πn

"）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

sin (x) = \frac{17320.508}{19999.99986 \dots} : x = arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn, x = π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

sin (x) = \frac{17320.508}{19999.99986 \dots}

使用反三角函数性质

sin (x) = \frac{17320.508}{19999.99986 \dots}

sin (x) = \frac{17320.508}{19999.99986 \dots}

的通解

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

x = arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn, x = π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

x = arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn, x = π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn, x = π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

将解代入原方程进行验证

将它们代入

- 50 cos (x) - 86.60254 \dots sin (x) = - 50

检验解是否符合

去除与方程不符的解。

检验

2 πn

的解:

真

2 πn

代入

n = 1

2 π 1

对于

- 50 cos (x) - 86.60254 \dots sin (x) = - 50

代入

x = 2 π 1

- 50 cos (2 π 1) - 86.60254 \dots sin (2 π 1) = - 50

整理后得

- 50 = - 50

\Rightarrow 真

检验

π + 2 πn

的解:

假

π + 2 πn

代入

n = 1

π + 2 π 1

对于

- 50 cos (x) - 86.60254 \dots sin (x) = - 50

代入

x = π + 2 π 1

- 50 cos (π + 2 π 1) - 86.60254 \dots sin (π + 2 π 1) = - 50

整理后得

50 = - 50

\Rightarrow 假

检验

arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

的解:

假

arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

代入

n = 1

arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1

对于

- 50 cos (x) - 86.60254 \dots sin (x) = - 50

代入

x = arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1

- 50 cos (arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1) - 86.60254 \dots sin (arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1) = - 50

整理后得

- 99.99999 \dots = - 50

\Rightarrow 假

检验

π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

的解:

真

π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

代入

n = 1

π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1

对于

- 50 cos (x) - 86.60254 \dots sin (x) = - 50

代入

x = π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1

- 50 cos (π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1) - 86.60254 \dots sin (π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 π 1) = - 50

整理后得

- 50 = - 50

\Rightarrow 真

x = 2 πn, x = π - arcsin (\frac{17320.508}{19999.99986 \dots}) + 2 πn

以小数形式表示解

x = 2 πn, x = π - 1.04719 \dots + 2 πn

作图

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