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受欢迎的三角函数 >

5sin(x)-5cos(x)=2

解答

5 sin (x) - 5 cos (x) = 2

解答

x = - 2.64295 \dots + 2 πn, x = 1.07215 \dots + 2 πn

+1

度数

x = - 151.42994 \dots^{\circ} + 36 0^{\circ} n, x = 61.42994 \dots^{\circ} + 36 0^{\circ} n

求解步骤

5 sin (x) - 5 cos (x) = 2

两边加上

5 cos (x)

5 sin (x) = 2 + 5 cos (x)

两边进行平方

(5 sin (x))^{2} = (2 + 5 cos (x))^{2}

两边减去

(2 + 5 cos (x))^{2}

25 sin^{2} (x) - 4 - 20 cos (x) - 25 cos^{2} (x) = 0

使用三角恒等式改写

- 4 - 20 cos (x) - 25 cos^{2} (x) + 25 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)

= - 4 - 20 cos (x) - 25 cos^{2} (x) + 25 (1 - cos^{2} (x))

化简

- 4 - 20 cos (x) - 25 cos^{2} (x) + 25 (1 - cos^{2} (x)) : - 50 cos^{2} (x) - 20 cos (x) + 21

- 4 - 20 cos (x) - 25 cos^{2} (x) + 25 (1 - cos^{2} (x))

乘开

25 (1 - cos^{2} (x)) : 25 - 25 cos^{2} (x)

25 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 25, b = 1, c = cos^{2} (x)

= 25 \cdot 1 - 25 cos^{2} (x)

数字相乘:

25 \cdot 1 = 25

= 25 - 25 cos^{2} (x)

= - 4 - 20 cos (x) - 25 cos^{2} (x) + 25 - 25 cos^{2} (x)

化简

- 4 - 20 cos (x) - 25 cos^{2} (x) + 25 - 25 cos^{2} (x) : - 50 cos^{2} (x) - 20 cos (x) + 21

- 4 - 20 cos (x) - 25 cos^{2} (x) + 25 - 25 cos^{2} (x)

对同类项分组

= - 20 cos (x) - 25 cos^{2} (x) - 25 cos^{2} (x) - 4 + 25

同类项相加:

- 25 cos^{2} (x) - 25 cos^{2} (x) = - 50 cos^{2} (x)

= - 20 cos (x) - 50 cos^{2} (x) - 4 + 25

数字相加/相减:

- 4 + 25 = 21

= - 50 cos^{2} (x) - 20 cos (x) + 21

= - 50 cos^{2} (x) - 20 cos (x) + 21

= - 50 cos^{2} (x) - 20 cos (x) + 21

21 - 20 cos (x) - 50 cos^{2} (x) = 0

用替代法求解

21 - 20 cos (x) - 50 cos^{2} (x) = 0

令

: cos (x) = u

21 - 20 u - 50 u^{2} = 0

21 - 20 u - 50 u^{2} = 0 : u = - \frac{2 + 46}{10}, u = \frac{46 - 2}{10}

21 - 20 u - 50 u^{2} = 0

改写成标准形式

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

- 50 u^{2} - 20 u + 21 = 0

使用求根公式求解

- 50 u^{2} - 20 u + 21 = 0

二次方程求根公式:

若

a = - 50, b = - 20, c = 21

u_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 ( - 50 ) \cdot 21}{2 ( - 50 )}

u_{1, 2} = \frac{- ( - 20 ) \pm ( - 20 ) ^{2} - 4 ( - 50 ) \cdot 21}{2 ( - 50 )}

(- 20)^{2} - 4 (- 50) \cdot 21 = 1046

(- 20)^{2} - 4 (- 50) \cdot 21

使用法则

- (- a) = a

= (- 20)^{2} + 4 \cdot 50 \cdot 21

使用指数法则:

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

若

n

是偶数

(- 20)^{2} = 2 0^{2}

= 2 0^{2} + 4 \cdot 50 \cdot 21

数字相乘:

4 \cdot 50 \cdot 21 = 4200

= 2 0^{2} + 4200

2 0^{2} = 400

= 400 + 4200

数字相加:

400 + 4200 = 4600

= 4600

4600

质因数分解:

2^{3} \cdot 5^{2} \cdot 23

4600

4600

除以

24600 = 2300 \cdot 2

= 2 \cdot 2300

2300

除以

22300 = 1150 \cdot 2

= 2 \cdot 2 \cdot 1150

1150

除以

21150 = 575 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 575

575

除以

5575 = 115 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 115

115

除以

5115 = 23 \cdot 5

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 23

2, 5, 23

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 23

= 2^{3} \cdot 5^{2} \cdot 23

= 2^{3} \cdot 5^{2} \cdot 23

使用指数法则:

a^{b + c} = a^{b} \cdot a^{c}

= 2^{2} \cdot 5^{2} \cdot 2 \cdot 23

使用根式运算法则:

n ab = n a n b

= 2^{2} 5^{2} 2 \cdot 23

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2 5^{2} 2 \cdot 23

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 2 \cdot 5 2 \cdot 23

整理后得

= 1046

u_{1, 2} = \frac{- ( - 20 ) \pm 10 46}{2 ( - 50 )}

将解分隔开

u_{1} = \frac{- ( - 20 ) + 10 46}{2 ( - 50 )}, u_{2} = \frac{- ( - 20 ) - 10 46}{2 ( - 50 )}

u = \frac{- ( - 20 ) + 10 46}{2 ( - 50 )} : - \frac{2 + 46}{10}

\frac{- ( - 20 ) + 10 46}{2 ( - 50 )}

去除括号:

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

= \frac{20 + 10 46}{- 2 \cdot 50}

数字相乘:

2 \cdot 50 = 100

= \frac{20 + 10 46}{- 100}

使用分式法则:

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

= - \frac{20 + 10 46}{100}

消掉

\frac{20 + 10 46}{100} : \frac{2 + 46}{10}

\frac{20 + 10 46}{100}

分解

20 + 1046 : 10 (2 + 46)

20 + 1046

改写为

= 10 \cdot 2 + 1046

因式分解出通项

10

= 10 (2 + 46)

= \frac{10 ( 2 + 46 )}{100}

约分:

10

= \frac{2 + 46}{10}

= - \frac{2 + 46}{10}

u = \frac{- ( - 20 ) - 10 46}{2 ( - 50 )} : \frac{46 - 2}{10}

\frac{- ( - 20 ) - 10 46}{2 ( - 50 )}

去除括号:

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

= \frac{20 - 10 46}{- 2 \cdot 50}

数字相乘:

2 \cdot 50 = 100

= \frac{20 - 10 46}{- 100}

使用分式法则:

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

20 - 1046 = - (1046 - 20)

= \frac{10 46 - 20}{100}

分解

1046 - 20 : 10 (46 - 2)

1046 - 20

改写为

= 1046 - 10 \cdot 2

因式分解出通项

10

= 10 (46 - 2)

= \frac{10 ( 46 - 2 )}{100}

约分:

10

= \frac{46 - 2}{10}

二次方程组的解是:

u = - \frac{2 + 46}{10}, u = \frac{46 - 2}{10}

u = cos (x)

代回

cos (x) = - \frac{2 + 46}{10}, cos (x) = \frac{46 - 2}{10}

cos (x) = - \frac{2 + 46}{10}, cos (x) = \frac{46 - 2}{10}

cos (x) = - \frac{2 + 46}{10} : x = arccos (- \frac{2 + 46}{10}) + 2 πn, x = - arccos (- \frac{2 + 46}{10}) + 2 πn

cos (x) = - \frac{2 + 46}{10}

使用反三角函数性质

cos (x) = - \frac{2 + 46}{10}

cos (x) = - \frac{2 + 46}{10}

的通解

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

x = arccos (- \frac{2 + 46}{10}) + 2 πn, x = - arccos (- \frac{2 + 46}{10}) + 2 πn

x = arccos (- \frac{2 + 46}{10}) + 2 πn, x = - arccos (- \frac{2 + 46}{10}) + 2 πn

cos (x) = \frac{46 - 2}{10} : x = arccos (\frac{46 - 2}{10}) + 2 πn, x = 2 π - arccos (\frac{46 - 2}{10}) + 2 πn

cos (x) = \frac{46 - 2}{10}

使用反三角函数性质

cos (x) = \frac{46 - 2}{10}

cos (x) = \frac{46 - 2}{10}

的通解

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

x = arccos (\frac{46 - 2}{10}) + 2 πn, x = 2 π - arccos (\frac{46 - 2}{10}) + 2 πn

x = arccos (\frac{46 - 2}{10}) + 2 πn, x = 2 π - arccos (\frac{46 - 2}{10}) + 2 πn

合并所有解

x = arccos (- \frac{2 + 46}{10}) + 2 πn, x = - arccos (- \frac{2 + 46}{10}) + 2 πn, x = arccos (\frac{46 - 2}{10}) + 2 πn, x = 2 π - arccos (\frac{46 - 2}{10}) + 2 πn

将解代入原方程进行验证

将它们代入

5 sin (x) - 5 cos (x) = 2

检验解是否符合

去除与方程不符的解。

检验

arccos (- \frac{2 + 46}{10}) + 2 πn

的解:

假

arccos (- \frac{2 + 46}{10}) + 2 πn

代入

n = 1

arccos (- \frac{2 + 46}{10}) + 2 π 1

对于

5 sin (x) - 5 cos (x) = 2

代入

x = arccos (- \frac{2 + 46}{10}) + 2 π 1

5 sin (arccos (- \frac{2 + 46}{10}) + 2 π 1) - 5 cos (arccos (- \frac{2 + 46}{10}) + 2 π 1) = 2

整理后得

6.78232 \dots = 2

\Rightarrow 假

检验

- arccos (- \frac{2 + 46}{10}) + 2 πn

的解:

真

- arccos (- \frac{2 + 46}{10}) + 2 πn

代入

n = 1

- arccos (- \frac{2 + 46}{10}) + 2 π 1

对于

5 sin (x) - 5 cos (x) = 2

代入

x = - arccos (- \frac{2 + 46}{10}) + 2 π 1

5 sin (- arccos (- \frac{2 + 46}{10}) + 2 π 1) - 5 cos (- arccos (- \frac{2 + 46}{10}) + 2 π 1) = 2

整理后得

2 = 2

\Rightarrow 真

检验

arccos (\frac{46 - 2}{10}) + 2 πn

的解:

真

arccos (\frac{46 - 2}{10}) + 2 πn

代入

n = 1

arccos (\frac{46 - 2}{10}) + 2 π 1

对于

5 sin (x) - 5 cos (x) = 2

代入

x = arccos (\frac{46 - 2}{10}) + 2 π 1

5 sin (arccos (\frac{46 - 2}{10}) + 2 π 1) - 5 cos (arccos (\frac{46 - 2}{10}) + 2 π 1) = 2

整理后得

2 = 2

\Rightarrow 真

检验

2 π - arccos (\frac{46 - 2}{10}) + 2 πn

的解:

假

2 π - arccos (\frac{46 - 2}{10}) + 2 πn

代入

n = 1

2 π - arccos (\frac{46 - 2}{10}) + 2 π 1

对于

5 sin (x) - 5 cos (x) = 2

代入

x = 2 π - arccos (\frac{46 - 2}{10}) + 2 π 1

5 sin (2 π - arccos (\frac{46 - 2}{10}) + 2 π 1) - 5 cos (2 π - arccos (\frac{46 - 2}{10}) + 2 π 1) = 2

整理后得

- 6.78232 \dots = 2

\Rightarrow 假

x = - arccos (- \frac{2 + 46}{10}) + 2 πn, x = arccos (\frac{46 - 2}{10}) + 2 πn

以小数形式表示解

x = - 2.64295 \dots + 2 πn, x = 1.07215 \dots + 2 πn

作图

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