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受欢迎的三角函数 >

90-70sin(x)-130cos(x)=0

解答

90 - 70 sin (x) - 130 cos (x) = 0

解答

x = 1.40923 \dots + 2 πn, x = - 0.42135 \dots + 2 πn

+1

度数

x = 80.74328 \dots^{\circ} + 36 0^{\circ} n, x = - 24.14177 \dots^{\circ} + 36 0^{\circ} n

求解步骤

90 - 70 sin (x) - 130 cos (x) = 0

两边加上

130 cos (x)

90 - 70 sin (x) = 130 cos (x)

两边进行平方

(90 - 70 sin (x))^{2} = (130 cos (x))^{2}

两边减去

(130 cos (x))^{2}

(90 - 70 sin (x))^{2} - 16900 cos^{2} (x) = 0

使用三角恒等式改写

(90 - 70 sin (x))^{2} - 16900 cos^{2} (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)

= (90 - 70 sin (x))^{2} - 16900 (1 - sin^{2} (x))

化简

(90 - 70 sin (x))^{2} - 16900 (1 - sin^{2} (x)) : 21800 sin^{2} (x) - 12600 sin (x) - 8800

(90 - 70 sin (x))^{2} - 16900 (1 - sin^{2} (x))

(90 - 70 sin (x))^{2} : 8100 - 12600 sin (x) + 4900 sin^{2} (x)

使用完全平方公式:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 90, b = 70 sin (x)

= 9 0^{2} - 2 \cdot 90 \cdot 70 sin (x) + (70 sin (x))^{2}

化简

9 0^{2} - 2 \cdot 90 \cdot 70 sin (x) + (70 sin (x))^{2} : 8100 - 12600 sin (x) + 4900 sin^{2} (x)

9 0^{2} - 2 \cdot 90 \cdot 70 sin (x) + (70 sin (x))^{2}

9 0^{2} = 8100

9 0^{2}

9 0^{2} = 8100

= 8100

2 \cdot 90 \cdot 70 sin (x) = 12600 sin (x)

2 \cdot 90 \cdot 70 sin (x)

数字相乘:

2 \cdot 90 \cdot 70 = 12600

= 12600 sin (x)

(70 sin (x))^{2} = 4900 sin^{2} (x)

(70 sin (x))^{2}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

= 7 0^{2} sin^{2} (x)

7 0^{2} = 4900

= 4900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 (1 - sin^{2} (x))

乘开

- 16900 (1 - sin^{2} (x)) : - 16900 + 16900 sin^{2} (x)

- 16900 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 16900, b = 1, c = sin^{2} (x)

= - 16900 \cdot 1 - (- 16900) sin^{2} (x)

使用加减运算法则

- (- a) = a

= - 16900 \cdot 1 + 16900 sin^{2} (x)

数字相乘:

16900 \cdot 1 = 16900

= - 16900 + 16900 sin^{2} (x)

= 8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 + 16900 sin^{2} (x)

化简

8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 + 16900 sin^{2} (x) : 21800 sin^{2} (x) - 12600 sin (x) - 8800

8100 - 12600 sin (x) + 4900 sin^{2} (x) - 16900 + 16900 sin^{2} (x)

对同类项分组

= - 12600 sin (x) + 4900 sin^{2} (x) + 16900 sin^{2} (x) + 8100 - 16900

同类项相加:

4900 sin^{2} (x) + 16900 sin^{2} (x) = 21800 sin^{2} (x)

= - 12600 sin (x) + 21800 sin^{2} (x) + 8100 - 16900

数字相加/相减:

8100 - 16900 = - 8800

= 21800 sin^{2} (x) - 12600 sin (x) - 8800

= 21800 sin^{2} (x) - 12600 sin (x) - 8800

= 21800 sin^{2} (x) - 12600 sin (x) - 8800

- 8800 - 12600 sin (x) + 21800 sin^{2} (x) = 0

用替代法求解

- 8800 - 12600 sin (x) + 21800 sin^{2} (x) = 0

令

: sin (x) = u

- 8800 - 12600 u + 21800 u^{2} = 0

- 8800 - 12600 u + 21800 u^{2} = 0 : u = \frac{63 + 13 137}{218}, u = \frac{63 - 13 137}{218}

- 8800 - 12600 u + 21800 u^{2} = 0

两边除以

21800

- \frac{8800}{21800} - \frac{12600 u}{21800} + \frac{21800 u ^{2}}{21800} = \frac{0}{21800}

改写成标准形式

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

u^{2} - \frac{63 u}{109} - \frac{44}{109} = 0

使用求根公式求解

u^{2} - \frac{63 u}{109} - \frac{44}{109} = 0

二次方程求根公式:

若

a = 1, b = - \frac{63}{109}, c = - \frac{44}{109}

u_{1, 2} = \frac{- ( - \frac{63}{109} ) \pm ( - \frac{63}{109} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{44}{109} )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - \frac{63}{109} ) \pm ( - \frac{63}{109} ) ^{2} - 4 \cdot 1 \cdot ( - \frac{44}{109} )}{2 \cdot 1}

(- \frac{63}{109})^{2} - 4 \cdot 1 \cdot (- \frac{44}{109}) = \frac{13 137}{109}

(- \frac{63}{109})^{2} - 4 \cdot 1 \cdot (- \frac{44}{109})

使用法则

- (- a) = a

= (- \frac{63}{109})^{2} + 4 \cdot 1 \cdot \frac{44}{109}

(- \frac{63}{109})^{2} = \frac{6 3 ^{2}}{10 9 ^{2}}

(- \frac{63}{109})^{2}

使用指数法则:

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

若

n

是偶数

(- \frac{63}{109})^{2} = (\frac{63}{109})^{2}

= (\frac{63}{109})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{6 3 ^{2}}{10 9 ^{2}}

4 \cdot 1 \cdot \frac{44}{109} = \frac{176}{109}

4 \cdot 1 \cdot \frac{44}{109}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 1 \cdot \frac{44 \cdot 4}{109}

数字相乘:

44 \cdot 4 = 176

= 1 \cdot \frac{176}{109}

乘以:

1 \cdot \frac{176}{109} = \frac{176}{109}

= \frac{176}{109}

= \frac{6 3 ^{2}}{10 9 ^{2}} + \frac{176}{109}

\frac{6 3 ^{2}}{10 9 ^{2}} = \frac{3969}{11881}

\frac{6 3 ^{2}}{10 9 ^{2}}

6 3^{2} = 3969

= \frac{3969}{10 9 ^{2}}

10 9^{2} = 11881

= \frac{3969}{11881}

= \frac{3969}{11881} + \frac{176}{109}

化简

\frac{3969}{11881} + \frac{176}{109} : \frac{23153}{11881}

\frac{3969}{11881} + \frac{176}{109}

11881, 109

的最小公倍数:

11881

11881, 109

最小公倍数 (LCM)

11881

质因数分解:

109 \cdot 109

11881

11881

除以

10911881 = 109 \cdot 109

= 109 \cdot 109

109

质因数分解:

109

109

109

是质数，因此无法因数分解

= 109

将每个因子乘以它在

11881

或

109

中出现的最多次数

= 109 \cdot 109

数字相乘:

109 \cdot 109 = 11881

= 11881

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

11881

对于

\frac{176}{109} :

将分母和分子乘以

109

\frac{176}{109} = \frac{176 \cdot 109}{109 \cdot 109} = \frac{19184}{11881}

= \frac{3969}{11881} + \frac{19184}{11881}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{3969 + 19184}{11881}

数字相加:

3969 + 19184 = 23153

= \frac{23153}{11881}

= \frac{23153}{11881}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{23153}{11881}

11881 = 109

11881

因式分解数字:

11881 = 10 9^{2}

= 10 9^{2}

使用根式运算法则:

n a^{n} = a

10 9^{2} = 109

= 109

= \frac{23153}{109}

23153 = 13137

23153

23153

质因数分解:

1 3^{2} \cdot 137

23153

23153

除以

1323153 = 1781 \cdot 13

= 13 \cdot 1781

1781

除以

131781 = 137 \cdot 13

= 13 \cdot 13 \cdot 137

13, 137

都是质数，因此无法进一步因数分解

= 13 \cdot 13 \cdot 137

= 1 3^{2} \cdot 137

= 1 3^{2} \cdot 137

使用根式运算法则:

n ab = n a n b

= 137 1 3^{2}

使用根式运算法则:

n a^{n} = a

1 3^{2} = 13

= 13137

= \frac{13 137}{109}

u_{1, 2} = \frac{- ( - \frac{63}{109} ) \pm \frac{13 137}{109}}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - \frac{63}{109} ) + \frac{13 137}{109}}{2 \cdot 1}, u_{2} = \frac{- ( - \frac{63}{109} ) - \frac{13 137}{109}}{2 \cdot 1}

u = \frac{- ( - \frac{63}{109} ) + \frac{13 137}{109}}{2 \cdot 1} : \frac{63 + 13 137}{218}

\frac{- ( - \frac{63}{109} ) + \frac{13 137}{109}}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{\frac{63}{109} + \frac{13 137}{109}}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{\frac{63}{109} + \frac{13 137}{109}}{2}

合并分式

\frac{63}{109} + \frac{13 137}{109} : \frac{63 + 13 137}{109}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{63 + 13 137}{109}

= \frac{\frac{63 + 13 137}{109}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{63 + 13 137}{109 \cdot 2}

数字相乘:

109 \cdot 2 = 218

= \frac{63 + 13 137}{218}

u = \frac{- ( - \frac{63}{109} ) - \frac{13 137}{109}}{2 \cdot 1} : \frac{63 - 13 137}{218}

\frac{- ( - \frac{63}{109} ) - \frac{13 137}{109}}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{\frac{63}{109} - \frac{13 137}{109}}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{\frac{63}{109} - \frac{13 137}{109}}{2}

合并分式

\frac{63}{109} - \frac{13 137}{109} : \frac{63 - 13 137}{109}

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{63 - 13 137}{109}

= \frac{\frac{63 - 13 137}{109}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{63 - 13 137}{109 \cdot 2}

数字相乘:

109 \cdot 2 = 218

= \frac{63 - 13 137}{218}

二次方程组的解是:

u = \frac{63 + 13 137}{218}, u = \frac{63 - 13 137}{218}

u = sin (x)

代回

sin (x) = \frac{63 + 13 137}{218}, sin (x) = \frac{63 - 13 137}{218}

sin (x) = \frac{63 + 13 137}{218}, sin (x) = \frac{63 - 13 137}{218}

sin (x) = \frac{63 + 13 137}{218} : x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

sin (x) = \frac{63 + 13 137}{218}

使用反三角函数性质

sin (x) = \frac{63 + 13 137}{218}

sin (x) = \frac{63 + 13 137}{218}

的通解

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

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

sin (x) = \frac{63 - 13 137}{218} : x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

sin (x) = \frac{63 - 13 137}{218}

使用反三角函数性质

sin (x) = \frac{63 - 13 137}{218}

sin (x) = \frac{63 - 13 137}{218}

的通解

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

x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

合并所有解

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = arcsin (\frac{63 - 13 137}{218}) + 2 πn, x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

将解代入原方程进行验证

将它们代入

90 - 70 sin (x) - 130 cos (x) = 0

检验解是否符合

去除与方程不符的解。

检验

arcsin (\frac{63 + 13 137}{218}) + 2 πn

的解:

真

arcsin (\frac{63 + 13 137}{218}) + 2 πn

代入

n = 1

arcsin (\frac{63 + 13 137}{218}) + 2 π 1

对于

90 - 70 sin (x) - 130 cos (x) = 0

代入

x = arcsin (\frac{63 + 13 137}{218}) + 2 π 1

90 - 70 sin (arcsin (\frac{63 + 13 137}{218}) + 2 π 1) - 130 cos (arcsin (\frac{63 + 13 137}{218}) + 2 π 1) = 0

整理后得

0 = 0

\Rightarrow 真

检验

π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

的解:

假

π - arcsin (\frac{63 + 13 137}{218}) + 2 πn

代入

n = 1

π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1

对于

90 - 70 sin (x) - 130 cos (x) = 0

代入

x = π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1

90 - 70 sin (π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1) - 130 cos (π - arcsin (\frac{63 + 13 137}{218}) + 2 π 1) = 0

整理后得

41.82314 \dots = 0

\Rightarrow 假

检验

arcsin (\frac{63 - 13 137}{218}) + 2 πn

的解:

真

arcsin (\frac{63 - 13 137}{218}) + 2 πn

代入

n = 1

arcsin (\frac{63 - 13 137}{218}) + 2 π 1

对于

90 - 70 sin (x) - 130 cos (x) = 0

代入

x = arcsin (\frac{63 - 13 137}{218}) + 2 π 1

90 - 70 sin (arcsin (\frac{63 - 13 137}{218}) + 2 π 1) - 130 cos (arcsin (\frac{63 - 13 137}{218}) + 2 π 1) = 0

整理后得

0 = 0

\Rightarrow 真

检验

π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

的解:

假

π + arcsin (- \frac{63 - 13 137}{218}) + 2 πn

代入

n = 1

π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1

对于

90 - 70 sin (x) - 130 cos (x) = 0

代入

x = π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1

90 - 70 sin (π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1) - 130 cos (π + arcsin (- \frac{63 - 13 137}{218}) + 2 π 1) = 0

整理后得

237.25942 \dots = 0

\Rightarrow 假

x = arcsin (\frac{63 + 13 137}{218}) + 2 πn, x = arcsin (\frac{63 - 13 137}{218}) + 2 πn

以小数形式表示解

x = 1.40923 \dots + 2 πn, x = - 0.42135 \dots + 2 πn

作图

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