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受欢迎的三角函数 >

sqrt(3)sin(x)-3cos(x)=sqrt(6)

解答

3 sin (x) - 3 cos (x) = 6

解答

x = - 2.87979 \dots + 2 πn, x = 1.83259 \dots + 2 πn

+1

度数

x = - 16 5^{\circ} + 36 0^{\circ} n, x = 10 5^{\circ} + 36 0^{\circ} n

求解步骤

3 sin (x) - 3 cos (x) = 6

两边加上

3 cos (x)

3 sin (x) = 6 + 3 cos (x)

两边进行平方

(3 sin (x))^{2} = (6 + 3 cos (x))^{2}

两边减去

(6 + 3 cos (x))^{2}

3 sin^{2} (x) - 6 - 66 cos (x) - 9 cos^{2} (x) = 0

使用三角恒等式改写

- 6 + 3 sin^{2} (x) - 9 cos^{2} (x) - 6 cos (x) 6

使用毕达哥拉斯恒等式:

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

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

= - 6 + 3 (1 - cos^{2} (x)) - 9 cos^{2} (x) - 6 cos (x) 6

化简

- 6 + 3 (1 - cos^{2} (x)) - 9 cos^{2} (x) - 6 cos (x) 6 : - 12 cos^{2} (x) - 66 cos (x) - 3

- 6 + 3 (1 - cos^{2} (x)) - 9 cos^{2} (x) - 6 cos (x) 6

= - 6 + 3 (1 - cos^{2} (x)) - 9 cos^{2} (x) - 66 cos (x)

乘开

3 (1 - cos^{2} (x)) : 3 - 3 cos^{2} (x)

3 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = cos^{2} (x)

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

数字相乘:

3 \cdot 1 = 3

= 3 - 3 cos^{2} (x)

= - 6 + 3 - 3 cos^{2} (x) - 9 cos^{2} (x) - 6 cos (x) 6

化简

- 6 + 3 - 3 cos^{2} (x) - 9 cos^{2} (x) - 6 cos (x) 6 : - 12 cos^{2} (x) - 66 cos (x) - 3

- 6 + 3 - 3 cos^{2} (x) - 9 cos^{2} (x) - 6 cos (x) 6

同类项相加:

- 3 cos^{2} (x) - 9 cos^{2} (x) = - 12 cos^{2} (x)

= - 6 + 3 - 12 cos^{2} (x) - 66 cos (x)

数字相加/相减:

- 6 + 3 = - 3

= - 12 cos^{2} (x) - 66 cos (x) - 3

= - 12 cos^{2} (x) - 66 cos (x) - 3

= - 12 cos^{2} (x) - 66 cos (x) - 3

- 3 - 12 cos^{2} (x) - 6 cos (x) 6 = 0

用替代法求解

- 3 - 12 cos^{2} (x) - 6 cos (x) 6 = 0

令

: cos (x) = u

- 3 - 12 u^{2} - 6 u 6 = 0

- 3 - 12 u^{2} - 6 u 6 = 0 : u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

- 3 - 12 u^{2} - 6 u 6 = 0

改写成标准形式

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

- 12 u^{2} - 66 u - 3 = 0

使用求根公式求解

- 12 u^{2} - 66 u - 3 = 0

二次方程求根公式:

若

a = - 12, b = - 66, c = - 3

u_{1, 2} = \frac{- ( - 6 6 ) \pm ( - 6 6 ) ^{2} - 4 ( - 12 ) ( - 3 )}{2 ( - 12 )}

u_{1, 2} = \frac{- ( - 6 6 ) \pm ( - 6 6 ) ^{2} - 4 ( - 12 ) ( - 3 )}{2 ( - 12 )}

(- 66)^{2} - 4 (- 12) (- 3) = 62

(- 66)^{2} - 4 (- 12) (- 3)

使用法则

- (- a) = a

= (- 66)^{2} - 4 \cdot 12 \cdot 3

(- 66)^{2} = 6^{3}

(- 66)^{2}

使用指数法则:

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

若

n

是偶数

(- 66)^{2} = (66)^{2}

= (66)^{2}

使用指数法则:

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

= 6^{2} (6)^{2}

(6)^{2} : 6

使用根式运算法则:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{2}

使用指数法则:

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

= 6^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

分式相乘:

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

= \frac{1 \cdot 2}{2}

约分:

2

= 1

= 6

= 6^{2} \cdot 6

使用指数法则:

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

6^{2} \cdot 6 = 6^{2 + 1}

= 6^{2 + 1}

数字相加:

2 + 1 = 3

= 6^{3}

4 \cdot 12 \cdot 3 = 144

4 \cdot 12 \cdot 3

数字相乘:

4 \cdot 12 \cdot 3 = 144

= 144

= 6^{3} - 144

6^{3} = 216

= 216 - 144

数字相减:

216 - 144 = 72

= 72

72

质因数分解:

2^{3} \cdot 3^{2}

72

72

除以

272 = 36 \cdot 2

= 2 \cdot 36

36

除以

236 = 18 \cdot 2

= 2 \cdot 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

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

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

使用指数法则:

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

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

使用根式运算法则:

n ab = n a n b

= 2 2^{2} 3^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 22 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 2 \cdot 32

整理后得

= 62

u_{1, 2} = \frac{- ( - 6 6 ) \pm 6 2}{2 ( - 12 )}

将解分隔开

u_{1} = \frac{- ( - 6 6 ) + 6 2}{2 ( - 12 )}, u_{2} = \frac{- ( - 6 6 ) - 6 2}{2 ( - 12 )}

u = \frac{- ( - 6 6 ) + 6 2}{2 ( - 12 )} : - \frac{6 + 2}{4}

\frac{- ( - 6 6 ) + 6 2}{2 ( - 12 )}

去除括号:

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

= \frac{6 6 + 6 2}{- 2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{6 6 + 6 2}{- 24}

使用分式法则:

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

= - \frac{6 6 + 6 2}{24}

消掉

\frac{6 6 + 6 2}{24} : \frac{6 + 2}{4}

\frac{6 6 + 6 2}{24}

因式分解出通项

6

= \frac{6 ( 6 + 2 )}{24}

约分:

6

= \frac{6 + 2}{4}

= - \frac{6 + 2}{4}

u = \frac{- ( - 6 6 ) - 6 2}{2 ( - 12 )} : - \frac{6 - 2}{4}

\frac{- ( - 6 6 ) - 6 2}{2 ( - 12 )}

去除括号:

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

= \frac{6 6 - 6 2}{- 2 \cdot 12}

数字相乘:

2 \cdot 12 = 24

= \frac{6 6 - 6 2}{- 24}

使用分式法则:

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

= - \frac{6 6 - 6 2}{24}

消掉

\frac{6 6 - 6 2}{24} : \frac{6 - 2}{4}

\frac{6 6 - 6 2}{24}

因式分解出通项

6

= \frac{6 ( 6 - 2 )}{24}

约分:

6

= \frac{6 - 2}{4}

= - \frac{6 - 2}{4}

二次方程组的解是:

u = - \frac{6 + 2}{4}, u = - \frac{6 - 2}{4}

u = cos (x)

代回

cos (x) = - \frac{6 + 2}{4}, cos (x) = - \frac{6 - 2}{4}

cos (x) = - \frac{6 + 2}{4}, cos (x) = - \frac{6 - 2}{4}

cos (x) = - \frac{6 + 2}{4} : x = arccos (- \frac{6 + 2}{4}) + 2 πn, x = - arccos (- \frac{6 + 2}{4}) + 2 πn

cos (x) = - \frac{6 + 2}{4}

使用反三角函数性质

cos (x) = - \frac{6 + 2}{4}

cos (x) = - \frac{6 + 2}{4}

的通解

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

x = arccos (- \frac{6 + 2}{4}) + 2 πn, x = - arccos (- \frac{6 + 2}{4}) + 2 πn

x = arccos (- \frac{6 + 2}{4}) + 2 πn, x = - arccos (- \frac{6 + 2}{4}) + 2 πn

cos (x) = - \frac{6 - 2}{4} : x = arccos (- \frac{6 - 2}{4}) + 2 πn, x = - arccos (- \frac{6 - 2}{4}) + 2 πn

cos (x) = - \frac{6 - 2}{4}

使用反三角函数性质

cos (x) = - \frac{6 - 2}{4}

cos (x) = - \frac{6 - 2}{4}

的通解

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

x = arccos (- \frac{6 - 2}{4}) + 2 πn, x = - arccos (- \frac{6 - 2}{4}) + 2 πn

x = arccos (- \frac{6 - 2}{4}) + 2 πn, x = - arccos (- \frac{6 - 2}{4}) + 2 πn

合并所有解

x = arccos (- \frac{6 + 2}{4}) + 2 πn, x = - arccos (- \frac{6 + 2}{4}) + 2 πn, x = arccos (- \frac{6 - 2}{4}) + 2 πn, x = - arccos (- \frac{6 - 2}{4}) + 2 πn

将解代入原方程进行验证

将它们代入

3 sin (x) - 3 cos (x) = 6

检验解是否符合

去除与方程不符的解。

检验

arccos (- \frac{6 + 2}{4}) + 2 πn

的解:

假

arccos (- \frac{6 + 2}{4}) + 2 πn

代入

n = 1

arccos (- \frac{6 + 2}{4}) + 2 π 1

对于

3 sin (x) - 3 cos (x) = 6

代入

x = arccos (- \frac{6 + 2}{4}) + 2 π 1

3 sin (arccos (- \frac{6 + 2}{4}) + 2 π 1) - 3 cos (arccos (- \frac{6 + 2}{4}) + 2 π 1) = 6

整理后得

3.34606 \dots = 2.44948 \dots

\Rightarrow 假

检验

- arccos (- \frac{6 + 2}{4}) + 2 πn

的解:

真

- arccos (- \frac{6 + 2}{4}) + 2 πn

代入

n = 1

- arccos (- \frac{6 + 2}{4}) + 2 π 1

对于

3 sin (x) - 3 cos (x) = 6

代入

x = - arccos (- \frac{6 + 2}{4}) + 2 π 1

3 sin (- arccos (- \frac{6 + 2}{4}) + 2 π 1) - 3 cos (- arccos (- \frac{6 + 2}{4}) + 2 π 1) = 6

整理后得

2.44948 \dots = 2.44948 \dots

\Rightarrow 真

检验

arccos (- \frac{6 - 2}{4}) + 2 πn

的解:

真

arccos (- \frac{6 - 2}{4}) + 2 πn

代入

n = 1

arccos (- \frac{6 - 2}{4}) + 2 π 1

对于

3 sin (x) - 3 cos (x) = 6

代入

x = arccos (- \frac{6 - 2}{4}) + 2 π 1

3 sin (arccos (- \frac{6 - 2}{4}) + 2 π 1) - 3 cos (arccos (- \frac{6 - 2}{4}) + 2 π 1) = 6

整理后得

2.44948 \dots = 2.44948 \dots

\Rightarrow 真

检验

- arccos (- \frac{6 - 2}{4}) + 2 πn

的解:

假

- arccos (- \frac{6 - 2}{4}) + 2 πn

代入

n = 1

- arccos (- \frac{6 - 2}{4}) + 2 π 1

对于

3 sin (x) - 3 cos (x) = 6

代入

x = - arccos (- \frac{6 - 2}{4}) + 2 π 1

3 sin (- arccos (- \frac{6 - 2}{4}) + 2 π 1) - 3 cos (- arccos (- \frac{6 - 2}{4}) + 2 π 1) = 6

整理后得

- 0.89657 \dots = 2.44948 \dots

\Rightarrow 假

x = - arccos (- \frac{6 + 2}{4}) + 2 πn, x = arccos (- \frac{6 - 2}{4}) + 2 πn

以小数形式表示解

x = - 2.87979 \dots + 2 πn, x = 1.83259 \dots + 2 πn

作图

流行的例子

sin(x)= 4/(sqrt(41))

sin (x) = \frac{4}{41}

sin (x) = - 1.1

cos (2 x) = 0.25

sin (x) = - 0.1

189.4=100tan^2(45+θ/2)

189.4 = 100 tan^{2} (4 5^{\circ} + \frac{θ}{2})