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受欢迎的三角函数 >

9.8sin(α)-1.96cos(α)=0.47

解答

9.8 \cdot sin (α) - 1.96 \cdot cos (α) = 0.47

解答

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

+1

度数

α = - 171.38555 \dots^{\circ} + 36 0^{\circ} n, α = 14.00542 \dots^{\circ} + 36 0^{\circ} n

求解步骤

9.8 sin (α) - 1.96 cos (α) = 0.47

两边加上

1.96 cos (α)

9.8 sin (α) = 0.47 + 1.96 cos (α)

两边进行平方

(9.8 sin (α))^{2} = (0.47 + 1.96 cos (α))^{2}

两边减去

(0.47 + 1.96 cos (α))^{2}

96.04 sin^{2} (α) - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) = 0

使用三角恒等式改写

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 sin^{2} (α)

使用毕达哥拉斯恒等式:

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

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

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

化简

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α)) : - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 (1 - cos^{2} (α))

乘开

96.04 (1 - cos^{2} (α)) : 96.04 - 96.04 cos^{2} (α)

96.04 (1 - cos^{2} (α))

使用分配律:

a (b - c) = ab - a c

a = 96.04, b = 1, c = cos^{2} (α)

= 96.04 \cdot 1 - 96.04 cos^{2} (α)

= 1 \cdot 96.04 - 96.04 cos^{2} (α)

数字相乘:

1 \cdot 96.04 = 96.04

= 96.04 - 96.04 cos^{2} (α)

= - 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

化简

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α) : - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

- 0.2209 - 1.8424 cos (α) - 3.8416 cos^{2} (α) + 96.04 - 96.04 cos^{2} (α)

对同类项分组

= - 1.8424 cos (α) - 3.8416 cos^{2} (α) - 96.04 cos^{2} (α) - 0.2209 + 96.04

同类项相加:

- 3.8416 cos^{2} (α) - 96.04 cos^{2} (α) = - 99.8816 cos^{2} (α)

= - 1.8424 cos (α) - 99.8816 cos^{2} (α) - 0.2209 + 96.04

数字相加/相减:

- 0.2209 + 96.04 = 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

= - 99.8816 cos^{2} (α) - 1.8424 cos (α) + 95.8191

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

用替代法求解

95.8191 - 1.8424 cos (α) - 99.8816 cos^{2} (α) = 0

令

: cos (α) = u

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

95.8191 - 1.8424 u - 99.8816 u^{2} = 0 : u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

95.8191 - 1.8424 u - 99.8816 u^{2} = 0

改写成标准形式

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

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

使用求根公式求解

- 99.8816 u^{2} - 1.8424 u + 95.8191 = 0

二次方程求根公式:

若

a = - 99.8816, b = - 1.8424, c = 95.8191

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

u_{1, 2} = \frac{- ( - 1.8424 ) \pm ( - 1.8424 ) ^{2} - 4 ( - 99.8816 ) \cdot 95.8191}{2 ( - 99.8816 )}

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191 = 38285.654512

(- 1.8424)^{2} - 4 (- 99.8816) \cdot 95.8191

使用法则

- (- a) = a

= (- 1.8424)^{2} + 4 \cdot 99.8816 \cdot 95.8191

使用指数法则:

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

若

n

是偶数

(- 1.8424)^{2} = 1.842 4^{2}

= 1.842 4^{2} + 4 \cdot 95.8191 \cdot 99.8816

数字相乘:

4 \cdot 99.8816 \cdot 95.8191 = 38282.26007 \dots

= 1.842 4^{2} + 38282.26007 \dots

1.842 4^{2} = 3.39443776

= 3.39443776 + 38282.26007 \dots

数字相加:

3.39443776 + 38282.26007 \dots = 38285.654512

= 38285.654512

u_{1, 2} = \frac{- ( - 1.8424 ) \pm 38285.654512}{2 ( - 99.8816 )}

将解分隔开

u_{1} = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}, u_{2} = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

u = \frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )} : - \frac{1.8424 + 38285.654512}{199.7632}

\frac{- ( - 1.8424 ) + 38285.654512}{2 ( - 99.8816 )}

去除括号:

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

= \frac{1.8424 + 38285.654512}{- 2 \cdot 99.8816}

数字相乘:

2 \cdot 99.8816 = 199.7632

= \frac{1.8424 + 38285.654512}{- 199.7632}

使用分式法则:

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

= - \frac{1.8424 + 38285.654512}{199.7632}

u = \frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )} : \frac{38285.654512 - 1.8424}{199.7632}

\frac{- ( - 1.8424 ) - 38285.654512}{2 ( - 99.8816 )}

去除括号:

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

= \frac{1.8424 - 38285.654512}{- 2 \cdot 99.8816}

数字相乘:

2 \cdot 99.8816 = 199.7632

= \frac{1.8424 - 38285.654512}{- 199.7632}

使用分式法则:

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

1.8424 - 38285.654512 = - (38285.654512 - 1.8424)

= \frac{38285.654512 - 1.8424}{199.7632}

二次方程组的解是:

u = - \frac{1.8424 + 38285.654512}{199.7632}, u = \frac{38285.654512 - 1.8424}{199.7632}

u = cos (α)

代回

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}, cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632} : α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

使用反三角函数性质

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

cos (α) = - \frac{1.8424 + 38285.654512}{199.7632}

的通解

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

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632} : α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

使用反三角函数性质

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

cos (α) = \frac{38285.654512 - 1.8424}{199.7632}

的通解

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

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

合并所有解

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn, α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

将解代入原方程进行验证

将它们代入

9.8 sin (α) - 1.96 cos (α) = 0.47

检验解是否符合

去除与方程不符的解。

检验

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

的解:

假

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

代入

n = 1

arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

对于

9.8 sin (α) - 1.96 cos (α) = 0.47

代入

α = arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

整理后得

3.40577 \dots = 0.47

\Rightarrow 假

检验

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

的解:

真

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn

代入

n = 1

- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

对于

9.8 sin (α) - 1.96 cos (α) = 0.47

代入

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1

9.8 sin (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) - 1.96 cos (- arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 π 1) = 0.47

整理后得

0.47 = 0.47

\Rightarrow 真

检验

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

的解:

真

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

代入

n = 1

arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

对于

9.8 sin (α) - 1.96 cos (α) = 0.47

代入

α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

整理后得

0.47 = 0.47

\Rightarrow 真

检验

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

的解:

假

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

代入

n = 1

2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

对于

9.8 sin (α) - 1.96 cos (α) = 0.47

代入

α = 2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1

9.8 sin (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) - 1.96 cos (2 π - arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 π 1) = 0.47

整理后得

- 4.27346 \dots = 0.47

\Rightarrow 假

α = - arccos (- \frac{1.8424 + 38285.654512}{199.7632}) + 2 πn, α = arccos (\frac{38285.654512 - 1.8424}{199.7632}) + 2 πn

以小数形式表示解

α = - 2.99124 \dots + 2 πn, α = 0.24444 \dots + 2 πn

作图

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