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受欢迎的三角函数 >

229cos(a)+14715sin(a)=3959.55

解答

229 cos (a) + 14715 sin (a) = 3959.55

解答

a = 0.25684 \dots + 2 πn, a = π - 0.28796 \dots + 2 πn

+1

度数

a = 14.71615 \dots^{\circ} + 36 0^{\circ} n, a = 163.50067 \dots^{\circ} + 36 0^{\circ} n

求解步骤

229 cos (a) + 14715 sin (a) = 3959.55

两边减去

14715 sin (a)

229 cos (a) = 3959.55 - 14715 sin (a)

两边进行平方

(229 cos (a))^{2} = (3959.55 - 14715 sin (a))^{2}

两边减去

(3959.55 - 14715 sin (a))^{2}

52441 cos^{2} (a) - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) = 0

使用三角恒等式改写

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 cos^{2} (a)

使用毕达哥拉斯恒等式:

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

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

= - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a))

化简

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a)) : 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a))

乘开

52441 (1 - sin^{2} (a)) : 52441 - 52441 sin^{2} (a)

52441 (1 - sin^{2} (a))

使用分配律:

a (b - c) = ab - a c

a = 52441, b = 1, c = sin^{2} (a)

= 52441 \cdot 1 - 52441 sin^{2} (a)

数字相乘:

52441 \cdot 1 = 52441

= 52441 - 52441 sin^{2} (a)

= - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a)

化简

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a) : 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a)

对同类项分组

= 116529556.5 sin (a) - 216531225 sin^{2} (a) - 52441 sin^{2} (a) - 15678036.2025 + 52441

同类项相加:

- 216531225 sin^{2} (a) - 52441 sin^{2} (a) = - 216583666 sin^{2} (a)

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15678036.2025 + 52441

数字相加/相减:

- 15678036.2025 + 52441 = - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15625595.2025 + 116529556.5 sin (a) - 216583666 sin^{2} (a) = 0

用替代法求解

- 15625595.2025 + 116529556.5 sin (a) - 216583666 sin^{2} (a) = 0

令

: sin (a) = u

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0 : u = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, u = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0

改写成标准形式

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

- 216583666 u^{2} + 116529556.5 u - 15625595.2025 = 0

使用求根公式求解

- 216583666 u^{2} + 116529556.5 u - 15625595.2025 = 0

二次方程求根公式:

若

a = - 216583666, b = 116529556.5, c = - 15625595.2025

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 4 ( - 216583666 ) ( - 15625595.2025 )}{2 ( - 216583666 )}

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 4 ( - 216583666 ) ( - 15625595.2025 )}{2 ( - 216583666 )}

116529556. 5^{2} - 4 (- 216583666) (- 15625595.2025) = 116529556. 5^{2} - 1.3537 E 16

116529556. 5^{2} - 4 (- 216583666) (- 15625595.2025)

使用法则

- (- a) = a

= 116529556. 5^{2} - 4 \cdot 216583666 \cdot 15625595.2025

数字相乘:

4 \cdot 216583666 \cdot 15625595.2025 = 1.3537 E 16

= 116529556. 5^{2} - 1.3537 E 16

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

将解分隔开

u_{1} = \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}, u_{2} = \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

u = \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )} : \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

\frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

去除括号:

(- a) = - a

= \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{- 2 \cdot 216583666}

数字相乘:

2 \cdot 216583666 = 433167332

= \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{- 433167332}

使用分式法则:

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

- 116529556.5 + 116529556. 5^{2} - 1.3537 E 16 = - (- 116529556. 5^{2} - 1.3537 E 16 + 116529556.5)

= \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

u = \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )} : \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

\frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

去除括号:

(- a) = - a

= \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{- 2 \cdot 216583666}

数字相乘:

2 \cdot 216583666 = 433167332

= \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{- 433167332}

使用分式法则:

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

- 116529556.5 - 116529556. 5^{2} - 1.3537 E 16 = - (116529556. 5^{2} - 1.3537 E 16 + 116529556.5)

= \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

二次方程组的解是:

u = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, u = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

u = sin (a)

代回

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332} : a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

使用反三角函数性质

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

的通解

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

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332} : a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

使用反三角函数性质

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

的通解

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

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

合并所有解

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

将解代入原方程进行验证

将它们代入

229 cos (a) + 14715 sin (a) = 3959.55

检验解是否符合

去除与方程不符的解。

检验

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

的解:

真

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

代入

n = 1

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

对于

229 cos (a) + 14715 sin (a) = 3959.55

代入

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

整理后得

3959.54999 \dots = 3959.55

\Rightarrow 真

检验

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

的解:

假

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

代入

n = 1

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

对于

229 cos (a) + 14715 sin (a) = 3959.55

代入

a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

整理后得

3516.57416 \dots = 3959.55

\Rightarrow 假

检验

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

的解:

假

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

代入

n = 1

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

对于

229 cos (a) + 14715 sin (a) = 3959.55

代入

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

整理后得

4398.69096 \dots = 3959.55

\Rightarrow 假

检验

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

的解:

真

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

代入

n = 1

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

对于

229 cos (a) + 14715 sin (a) = 3959.55

代入

a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

整理后得

3959.54999 \dots = 3959.55

\Rightarrow 真

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

以小数形式表示解

a = 0.25684 \dots + 2 πn, a = π - 0.28796 \dots + 2 πn

作图

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