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受欢迎的三角函数 >

sin(A)-0.6*cos(A)=(2.77)/(9.8)

解答

sin (A) - 0.6 \cdot cos (A) = \frac{2.77}{9.8}

解答

A = - 2.84598 \dots + 2 πn, A = 0.78523 \dots + 2 πn

+1

度数

A = - 163.06288 \dots^{\circ} + 36 0^{\circ} n, A = 44.99039 \dots^{\circ} + 36 0^{\circ} n

求解步骤

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

两边加上

0.6 cos (A)

sin (A) = 0.28265 \dots + 0.6 cos (A)

两边进行平方

sin^{2} (A) = (0.28265 \dots + 0.6 cos (A))^{2}

两边减去

(0.28265 \dots + 0.6 cos (A))^{2}

sin^{2} (A) - 0.07989 \dots - 0.33918 \dots cos (A) - 0.36 cos^{2} (A) = 0

使用三角恒等式改写

- 0.07989 \dots + sin^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A)

使用毕达哥拉斯恒等式:

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

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

= - 0.07989 \dots + 1 - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A)

化简

- 0.07989 \dots + 1 - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A) : - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) + 0.92010 \dots

- 0.07989 \dots + 1 - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A)

对同类项分组

= - cos^{2} (A) - 0.33918 \dots cos (A) - 0.36 cos^{2} (A) - 0.07989 \dots + 1

同类项相加:

- cos^{2} (A) - 0.36 cos^{2} (A) = - 1.36 cos^{2} (A)

= - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) - 0.07989 \dots + 1

数字相加/相减:

- 0.07989 \dots + 1 = 0.92010 \dots

= - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) + 0.92010 \dots

= - 1.36 cos^{2} (A) - 0.33918 \dots cos (A) + 0.92010 \dots

0.92010 \dots - 0.33918 \dots cos (A) - 1.36 cos^{2} (A) = 0

用替代法求解

0.92010 \dots - 0.33918 \dots cos (A) - 1.36 cos^{2} (A) = 0

令

: cos (A) = u

0.92010 \dots - 0.33918 \dots u - 1.36 u^{2} = 0

0.92010 \dots - 0.33918 \dots u - 1.36 u^{2} = 0 : u = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, u = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

0.92010 \dots - 0.33918 \dots u - 1.36 u^{2} = 0

改写成标准形式

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

- 1.36 u^{2} - 0.33918 \dots u + 0.92010 \dots = 0

使用求根公式求解

- 1.36 u^{2} - 0.33918 \dots u + 0.92010 \dots = 0

二次方程求根公式:

若

a = - 1.36, b = - 0.33918 \dots, c = 0.92010 \dots

u_{1, 2} = \frac{- ( - 0.33918 \dots ) \pm ( - 0.33918 \dots ) ^{2} - 4 ( - 1.36 ) \cdot 0.92010 \dots}{2 ( - 1.36 )}

u_{1, 2} = \frac{- ( - 0.33918 \dots ) \pm ( - 0.33918 \dots ) ^{2} - 4 ( - 1.36 ) \cdot 0.92010 \dots}{2 ( - 1.36 )}

(- 0.33918 \dots)^{2} - 4 (- 1.36) \cdot 0.92010 \dots = 5.12042 \dots

(- 0.33918 \dots)^{2} - 4 (- 1.36) \cdot 0.92010 \dots

使用法则

- (- a) = a

= (- 0.33918 \dots)^{2} + 4 \cdot 1.36 \cdot 0.92010 \dots

使用指数法则:

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

若

n

是偶数

(- 0.33918 \dots)^{2} = 0.33918 \dots^{2}

= 0.33918 \dots^{2} + 4 \cdot 0.92010 \dots \cdot 1.36

数字相乘:

4 \cdot 1.36 \cdot 0.92010 \dots = 5.00538 \dots

= 0.33918 \dots^{2} + 5.00538 \dots

0.33918 \dots^{2} = 0.11504 \dots

= 0.11504 \dots + 5.00538 \dots

数字相加:

0.11504 \dots + 5.00538 \dots = 5.12042 \dots

= 5.12042 \dots

u_{1, 2} = \frac{- ( - 0.33918 \dots ) \pm 5.12042 \dots}{2 ( - 1.36 )}

将解分隔开

u_{1} = \frac{- ( - 0.33918 \dots ) + 5.12042 \dots}{2 ( - 1.36 )}, u_{2} = \frac{- ( - 0.33918 \dots ) - 5.12042 \dots}{2 ( - 1.36 )}

u = \frac{- ( - 0.33918 \dots ) + 5.12042 \dots}{2 ( - 1.36 )} : - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

\frac{- ( - 0.33918 \dots ) + 5.12042 \dots}{2 ( - 1.36 )}

去除括号:

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

= \frac{0.33918 \dots + 5.12042 \dots}{- 2 \cdot 1.36}

数字相乘:

2 \cdot 1.36 = 2.72

= \frac{0.33918 \dots + 5.12042 \dots}{- 2.72}

使用分式法则:

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

= - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

u = \frac{- ( - 0.33918 \dots ) - 5.12042 \dots}{2 ( - 1.36 )} : \frac{5.12042 \dots - 0.33918 \dots}{2.72}

\frac{- ( - 0.33918 \dots ) - 5.12042 \dots}{2 ( - 1.36 )}

去除括号:

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

= \frac{0.33918 \dots - 5.12042 \dots}{- 2 \cdot 1.36}

数字相乘:

2 \cdot 1.36 = 2.72

= \frac{0.33918 \dots - 5.12042 \dots}{- 2.72}

使用分式法则:

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

0.33918 \dots - 5.12042 \dots = - (5.12042 \dots - 0.33918 \dots)

= \frac{5.12042 \dots - 0.33918 \dots}{2.72}

二次方程组的解是:

u = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, u = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

u = cos (A)

代回

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}, cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72} : A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

使用反三角函数性质

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

cos (A) = - \frac{0.33918 \dots + 5.12042 \dots}{2.72}

的通解

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

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72} : A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

使用反三角函数性质

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

cos (A) = \frac{5.12042 \dots - 0.33918 \dots}{2.72}

的通解

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

A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

合并所有解

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn, A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

将解代入原方程进行验证

将它们代入

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

检验解是否符合

去除与方程不符的解。

检验

arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

的解:

假

arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

代入

n = 1

arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

对于

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

代入

A = arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

sin (arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) - 0.6 cos (arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

整理后得

0.86529 \dots = 0.28265 \dots

\Rightarrow 假

检验

- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

的解:

真

- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn

代入

n = 1

- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

对于

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

代入

A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1

sin (- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) - 0.6 cos (- arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

整理后得

0.28265 \dots = 0.28265 \dots

\Rightarrow 真

检验

arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

的解:

真

arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

代入

n = 1

arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

对于

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

代入

A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

sin (arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) - 0.6 cos (arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

整理后得

0.28265 \dots = 0.28265 \dots

\Rightarrow 真

检验

2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

的解:

假

2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

代入

n = 1

2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

对于

sin (A) - 0.6 cos (A) = \frac{2.77}{9.8}

代入

A = 2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1

sin (2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) - 0.6 cos (2 π - arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 π 1) = \frac{2.77}{9.8}

整理后得

- 1.13132 \dots = 0.28265 \dots

\Rightarrow 假

A = - arccos (- \frac{0.33918 \dots + 5.12042 \dots}{2.72}) + 2 πn, A = arccos (\frac{5.12042 \dots - 0.33918 \dots}{2.72}) + 2 πn

以小数形式表示解

A = - 2.84598 \dots + 2 πn, A = 0.78523 \dots + 2 πn

作图

流行的例子

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2cos(x)*tan(x)-1=0

2 cos (x) \cdot tan (x) - 1 = 0

sin (x) = - 0.465

cos (x) = \frac{10}{17}

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