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受欢迎的三角函数 >

6=50sin(x)-15cos(x),0<x< pi/2

解答

6 = 50 sin (x) - 15 cos (x), 0 < x < \frac{π}{2}

解答

x = 0.40665 \dots

+1

度数

x = 23.29935 \dots^{\circ}

求解步骤

6 = 50 sin (x) - 15 cos (x), 0 < x < \frac{π}{2}

两边加上

15 cos (x)

50 sin (x) = 6 + 15 cos (x)

两边进行平方

(50 sin (x))^{2} = (6 + 15 cos (x))^{2}

两边减去

(6 + 15 cos (x))^{2}

2500 sin^{2} (x) - 36 - 180 cos (x) - 225 cos^{2} (x) = 0

使用三角恒等式改写

- 36 - 180 cos (x) - 225 cos^{2} (x) + 2500 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)

= - 36 - 180 cos (x) - 225 cos^{2} (x) + 2500 (1 - cos^{2} (x))

化简

- 36 - 180 cos (x) - 225 cos^{2} (x) + 2500 (1 - cos^{2} (x)) : - 2725 cos^{2} (x) - 180 cos (x) + 2464

- 36 - 180 cos (x) - 225 cos^{2} (x) + 2500 (1 - cos^{2} (x))

乘开

2500 (1 - cos^{2} (x)) : 2500 - 2500 cos^{2} (x)

2500 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 2500, b = 1, c = cos^{2} (x)

= 2500 \cdot 1 - 2500 cos^{2} (x)

数字相乘:

2500 \cdot 1 = 2500

= 2500 - 2500 cos^{2} (x)

= - 36 - 180 cos (x) - 225 cos^{2} (x) + 2500 - 2500 cos^{2} (x)

化简

- 36 - 180 cos (x) - 225 cos^{2} (x) + 2500 - 2500 cos^{2} (x) : - 2725 cos^{2} (x) - 180 cos (x) + 2464

- 36 - 180 cos (x) - 225 cos^{2} (x) + 2500 - 2500 cos^{2} (x)

对同类项分组

= - 180 cos (x) - 225 cos^{2} (x) - 2500 cos^{2} (x) - 36 + 2500

同类项相加:

- 225 cos^{2} (x) - 2500 cos^{2} (x) = - 2725 cos^{2} (x)

= - 180 cos (x) - 2725 cos^{2} (x) - 36 + 2500

数字相加/相减:

- 36 + 2500 = 2464

= - 2725 cos^{2} (x) - 180 cos (x) + 2464

= - 2725 cos^{2} (x) - 180 cos (x) + 2464

= - 2725 cos^{2} (x) - 180 cos (x) + 2464

2464 - 180 cos (x) - 2725 cos^{2} (x) = 0

用替代法求解

2464 - 180 cos (x) - 2725 cos^{2} (x) = 0

令

: cos (x) = u

2464 - 180 u - 2725 u^{2} = 0

2464 - 180 u - 2725 u^{2} = 0 : u = - \frac{2 ( 9 + 5 2689 )}{545}, u = \frac{2 ( 5 2689 - 9 )}{545}

2464 - 180 u - 2725 u^{2} = 0

改写成标准形式

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

- 2725 u^{2} - 180 u + 2464 = 0

使用求根公式求解

- 2725 u^{2} - 180 u + 2464 = 0

二次方程求根公式:

若

a = - 2725, b = - 180, c = 2464

u_{1, 2} = \frac{- ( - 180 ) \pm ( - 180 ) ^{2} - 4 ( - 2725 ) \cdot 2464}{2 ( - 2725 )}

u_{1, 2} = \frac{- ( - 180 ) \pm ( - 180 ) ^{2} - 4 ( - 2725 ) \cdot 2464}{2 ( - 2725 )}

(- 180)^{2} - 4 (- 2725) \cdot 2464 = 1002689

(- 180)^{2} - 4 (- 2725) \cdot 2464

使用法则

- (- a) = a

= (- 180)^{2} + 4 \cdot 2725 \cdot 2464

使用指数法则:

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

若

n

是偶数

(- 180)^{2} = 18 0^{2}

= 18 0^{2} + 4 \cdot 2725 \cdot 2464

数字相乘:

4 \cdot 2725 \cdot 2464 = 26857600

= 18 0^{2} + 26857600

18 0^{2} = 32400

= 32400 + 26857600

数字相加:

32400 + 26857600 = 26890000

= 26890000

26890000

质因数分解:

2^{4} \cdot 5^{4} \cdot 2689

26890000

= 2^{4} \cdot 5^{4} \cdot 2689

使用根式运算法则:

n ab = n a n b

= 2689 2^{4} 5^{4}

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}}

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 2689 5^{4}

使用根式运算法则:

n a^{m} = a^{\frac{m}{n}}

5^{4} = 5^{\frac{4}{2}} = 5^{2}

= 2^{2} \cdot 5^{2} 2689

整理后得

= 1002689

u_{1, 2} = \frac{- ( - 180 ) \pm 100 2689}{2 ( - 2725 )}

将解分隔开

u_{1} = \frac{- ( - 180 ) + 100 2689}{2 ( - 2725 )}, u_{2} = \frac{- ( - 180 ) - 100 2689}{2 ( - 2725 )}

u = \frac{- ( - 180 ) + 100 2689}{2 ( - 2725 )} : - \frac{2 ( 9 + 5 2689 )}{545}

\frac{- ( - 180 ) + 100 2689}{2 ( - 2725 )}

去除括号:

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

= \frac{180 + 100 2689}{- 2 \cdot 2725}

数字相乘:

2 \cdot 2725 = 5450

= \frac{180 + 100 2689}{- 5450}

使用分式法则:

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

= - \frac{180 + 100 2689}{5450}

消掉

\frac{180 + 100 2689}{5450} : \frac{2 ( 9 + 5 2689 )}{545}

\frac{180 + 100 2689}{5450}

分解

180 + 1002689 : 20 (9 + 52689)

180 + 1002689

改写为

= 20 \cdot 9 + 20 \cdot 52689

因式分解出通项

20

= 20 (9 + 52689)

= \frac{20 ( 9 + 5 2689 )}{5450}

约分:

10

= \frac{2 ( 9 + 5 2689 )}{545}

= - \frac{2 ( 9 + 5 2689 )}{545}

u = \frac{- ( - 180 ) - 100 2689}{2 ( - 2725 )} : \frac{2 ( 5 2689 - 9 )}{545}

\frac{- ( - 180 ) - 100 2689}{2 ( - 2725 )}

去除括号:

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

= \frac{180 - 100 2689}{- 2 \cdot 2725}

数字相乘:

2 \cdot 2725 = 5450

= \frac{180 - 100 2689}{- 5450}

使用分式法则:

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

180 - 1002689 = - (1002689 - 180)

= \frac{100 2689 - 180}{5450}

分解

1002689 - 180 : 20 (52689 - 9)

1002689 - 180

改写为

= 20 \cdot 52689 - 20 \cdot 9

因式分解出通项

20

= 20 (52689 - 9)

= \frac{20 ( 5 2689 - 9 )}{5450}

约分:

10

= \frac{2 ( 5 2689 - 9 )}{545}

二次方程组的解是:

u = - \frac{2 ( 9 + 5 2689 )}{545}, u = \frac{2 ( 5 2689 - 9 )}{545}

u = cos (x)

代回

cos (x) = - \frac{2 ( 9 + 5 2689 )}{545}, cos (x) = \frac{2 ( 5 2689 - 9 )}{545}

cos (x) = - \frac{2 ( 9 + 5 2689 )}{545}, cos (x) = \frac{2 ( 5 2689 - 9 )}{545}

cos (x) = - \frac{2 ( 9 + 5 2689 )}{545}, 0 < x < \frac{π}{2} :

无解

cos (x) = - \frac{2 ( 9 + 5 2689 )}{545}, 0 < x < \frac{π}{2}

使用反三角函数性质

cos (x) = - \frac{2 ( 9 + 5 2689 )}{545}

cos (x) = - \frac{2 ( 9 + 5 2689 )}{545}

的通解

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

x = arccos (- \frac{2 ( 9 + 5 2689 )}{545}) + 2 πn, x = - arccos (- \frac{2 ( 9 + 5 2689 )}{545}) + 2 πn

x = arccos (- \frac{2 ( 9 + 5 2689 )}{545}) + 2 πn, x = - arccos (- \frac{2 ( 9 + 5 2689 )}{545}) + 2 πn

在

0 < x < \frac{π}{2}

范围内的解

无解

cos (x) = \frac{2 ( 5 2689 - 9 )}{545}, 0 < x < \frac{π}{2} : x = arccos (\frac{2 ( 5 2689 - 9 )}{545})

cos (x) = \frac{2 ( 5 2689 - 9 )}{545}, 0 < x < \frac{π}{2}

使用反三角函数性质

cos (x) = \frac{2 ( 5 2689 - 9 )}{545}

cos (x) = \frac{2 ( 5 2689 - 9 )}{545}

的通解

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

x = arccos (\frac{2 ( 5 2689 - 9 )}{545}) + 2 πn, x = 2 π - arccos (\frac{2 ( 5 2689 - 9 )}{545}) + 2 πn

x = arccos (\frac{2 ( 5 2689 - 9 )}{545}) + 2 πn, x = 2 π - arccos (\frac{2 ( 5 2689 - 9 )}{545}) + 2 πn

在

0 < x < \frac{π}{2}

范围内的解

x = arccos (\frac{2 ( 5 2689 - 9 )}{545})

合并所有解

x = arccos (\frac{2 ( 5 2689 - 9 )}{545})

将解代入原方程进行验证

将它们代入

50 sin (x) - 15 cos (x) = 6

检验解是否符合

去除与方程不符的解。

检验

arccos (\frac{2 ( 5 2689 - 9 )}{545})

的解:

真

arccos (\frac{2 ( 5 2689 - 9 )}{545})

代入

n = 1

arccos (\frac{2 ( 5 2689 - 9 )}{545})

对于

50 sin (x) - 15 cos (x) = 6

代入

x = arccos (\frac{2 ( 5 2689 - 9 )}{545})

50 sin (arccos (\frac{2 ( 5 2689 - 9 )}{545})) - 15 cos (arccos (\frac{2 ( 5 2689 - 9 )}{545})) = 6

整理后得

6 = 6

\Rightarrow 真

x = arccos (\frac{2 ( 5 2689 - 9 )}{545})

以小数形式表示解

x = 0.40665 \dots

作图

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