zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

9.8sin(x)-1.96cos(x)=8.54

解答

9.8 sin (x) - 1.96 cos (x) = 8.54

解答

x = 2.31438 \dots + 2 πn, x = 1.22199 \dots + 2 πn

+1

度数

x = 132.60472 \dots^{\circ} + 36 0^{\circ} n, x = 70.01513 \dots^{\circ} + 36 0^{\circ} n

求解步骤

9.8 sin (x) - 1.96 cos (x) = 8.54

两边加上

1.96 cos (x)

9.8 sin (x) = 8.54 + 1.96 cos (x)

两边进行平方

(9.8 sin (x))^{2} = (8.54 + 1.96 cos (x))^{2}

两边减去

(8.54 + 1.96 cos (x))^{2}

96.04 sin^{2} (x) - 72.9316 - 33.4768 cos (x) - 3.8416 cos^{2} (x) = 0

使用三角恒等式改写

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 sin^{2} (x)

使用毕达哥拉斯恒等式:

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

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

= - 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 (1 - cos^{2} (x))

化简

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 (1 - cos^{2} (x)) : - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 (1 - cos^{2} (x))

乘开

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

96.04 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 96.04, b = 1, c = cos^{2} (x)

= 96.04 \cdot 1 - 96.04 cos^{2} (x)

= 1 \cdot 96.04 - 96.04 cos^{2} (x)

数字相乘:

1 \cdot 96.04 = 96.04

= 96.04 - 96.04 cos^{2} (x)

= - 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 - 96.04 cos^{2} (x)

化简

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 - 96.04 cos^{2} (x) : - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

- 72.9316 - 3.8416 cos^{2} (x) - 33.4768 cos (x) + 96.04 - 96.04 cos^{2} (x)

对同类项分组

= - 3.8416 cos^{2} (x) - 33.4768 cos (x) - 96.04 cos^{2} (x) - 72.9316 + 96.04

同类项相加:

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

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) - 72.9316 + 96.04

数字相加/相减:

- 72.9316 + 96.04 = 23.1084

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

= - 99.8816 cos^{2} (x) - 33.4768 cos (x) + 23.1084

23.1084 - 33.4768 cos (x) - 99.8816 cos^{2} (x) = 0

用替代法求解

23.1084 - 33.4768 cos (x) - 99.8816 cos^{2} (x) = 0

令

: cos (x) = u

23.1084 - 33.4768 u - 99.8816 u^{2} = 0

23.1084 - 33.4768 u - 99.8816 u^{2} = 0 : u = - \frac{33.4768 + 10353.112}{199.7632}, u = \frac{10353.112 - 33.4768}{199.7632}

23.1084 - 33.4768 u - 99.8816 u^{2} = 0

改写成标准形式

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

- 99.8816 u^{2} - 33.4768 u + 23.1084 = 0

使用求根公式求解

- 99.8816 u^{2} - 33.4768 u + 23.1084 = 0

二次方程求根公式:

若

a = - 99.8816, b = - 33.4768, c = 23.1084

u_{1, 2} = \frac{- ( - 33.4768 ) \pm ( - 33.4768 ) ^{2} - 4 ( - 99.8816 ) \cdot 23.1084}{2 ( - 99.8816 )}

u_{1, 2} = \frac{- ( - 33.4768 ) \pm ( - 33.4768 ) ^{2} - 4 ( - 99.8816 ) \cdot 23.1084}{2 ( - 99.8816 )}

(- 33.4768)^{2} - 4 (- 99.8816) \cdot 23.1084 = 10353.112

(- 33.4768)^{2} - 4 (- 99.8816) \cdot 23.1084

使用法则

- (- a) = a

= (- 33.4768)^{2} + 4 \cdot 99.8816 \cdot 23.1084

使用指数法则:

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

若

n

是偶数

(- 33.4768)^{2} = 33.476 8^{2}

= 33.476 8^{2} + 4 \cdot 23.1084 \cdot 99.8816

数字相乘:

4 \cdot 99.8816 \cdot 23.1084 = 9232.41586 \dots

= 33.476 8^{2} + 9232.41586 \dots

33.476 8^{2} = 1120.69613 \dots

= 1120.69613 \dots + 9232.41586 \dots

数字相加:

1120.69613 \dots + 9232.41586 \dots = 10353.112

= 10353.112

u_{1, 2} = \frac{- ( - 33.4768 ) \pm 10353.112}{2 ( - 99.8816 )}

将解分隔开

u_{1} = \frac{- ( - 33.4768 ) + 10353.112}{2 ( - 99.8816 )}, u_{2} = \frac{- ( - 33.4768 ) - 10353.112}{2 ( - 99.8816 )}

u = \frac{- ( - 33.4768 ) + 10353.112}{2 ( - 99.8816 )} : - \frac{33.4768 + 10353.112}{199.7632}

\frac{- ( - 33.4768 ) + 10353.112}{2 ( - 99.8816 )}

去除括号:

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

= \frac{33.4768 + 10353.112}{- 2 \cdot 99.8816}

数字相乘:

2 \cdot 99.8816 = 199.7632

= \frac{33.4768 + 10353.112}{- 199.7632}

使用分式法则:

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

= - \frac{33.4768 + 10353.112}{199.7632}

u = \frac{- ( - 33.4768 ) - 10353.112}{2 ( - 99.8816 )} : \frac{10353.112 - 33.4768}{199.7632}

\frac{- ( - 33.4768 ) - 10353.112}{2 ( - 99.8816 )}

去除括号:

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

= \frac{33.4768 - 10353.112}{- 2 \cdot 99.8816}

数字相乘:

2 \cdot 99.8816 = 199.7632

= \frac{33.4768 - 10353.112}{- 199.7632}

使用分式法则:

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

33.4768 - 10353.112 = - (10353.112 - 33.4768)

= \frac{10353.112 - 33.4768}{199.7632}

二次方程组的解是:

u = - \frac{33.4768 + 10353.112}{199.7632}, u = \frac{10353.112 - 33.4768}{199.7632}

u = cos (x)

代回

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}, cos (x) = \frac{10353.112 - 33.4768}{199.7632}

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}, cos (x) = \frac{10353.112 - 33.4768}{199.7632}

cos (x) = - \frac{33.4768 + 10353.112}{199.7632} : x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}

使用反三角函数性质

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}

cos (x) = - \frac{33.4768 + 10353.112}{199.7632}

的通解

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

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

cos (x) = \frac{10353.112 - 33.4768}{199.7632} : x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

cos (x) = \frac{10353.112 - 33.4768}{199.7632}

使用反三角函数性质

cos (x) = \frac{10353.112 - 33.4768}{199.7632}

cos (x) = \frac{10353.112 - 33.4768}{199.7632}

的通解

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

x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

合并所有解

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn, x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

将解代入原方程进行验证

将它们代入

9.8 sin (x) - 1.96 cos (x) = 8.54

检验解是否符合

去除与方程不符的解。

检验

arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

的解:

真

arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

代入

n = 1

arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

对于

9.8 sin (x) - 1.96 cos (x) = 8.54

代入

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

9.8 sin (arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) - 1.96 cos (arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) = 8.54

整理后得

8.54 = 8.54

\Rightarrow 真

检验

- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

的解:

假

- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn

代入

n = 1

- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

对于

9.8 sin (x) - 1.96 cos (x) = 8.54

代入

x = - arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1

9.8 sin (- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) - 1.96 cos (- arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 π 1) = 8.54

整理后得

- 5.88640 \dots = 8.54

\Rightarrow 假

检验

arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

的解:

真

arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

代入

n = 1

arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

对于

9.8 sin (x) - 1.96 cos (x) = 8.54

代入

x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

9.8 sin (arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) - 1.96 cos (arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) = 8.54

整理后得

8.54 = 8.54

\Rightarrow 真

检验

2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

的解:

假

2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

代入

n = 1

2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

对于

9.8 sin (x) - 1.96 cos (x) = 8.54

代入

x = 2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1

9.8 sin (2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) - 1.96 cos (2 π - arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 π 1) = 8.54

整理后得

- 9.87974 \dots = 8.54

\Rightarrow 假

x = arccos (- \frac{33.4768 + 10353.112}{199.7632}) + 2 πn, x = arccos (\frac{10353.112 - 33.4768}{199.7632}) + 2 πn

以小数形式表示解

x = 2.31438 \dots + 2 πn, x = 1.22199 \dots + 2 πn

作图

流行的例子

3/(cot(x))=8-5tan(x)

\frac{3}{cot ( x )} = 8 - 5 tan (x)

sin(θ)= 3/8 ,sin(2θ)

sin (θ) = \frac{3}{8}, sin (2 θ)

2(1-cos^2(x))= 3/2

2 (1 - cos^{2} (x)) = \frac{3}{2}

cos(3θ)=-(sqrt(2))/2

cos (3 θ) = - \frac{2}{2}

25*sin(x)-1.5*cos(x)=20

25 \cdot sin (x) - 1.5 \cdot cos (x) = 20