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受欢迎的三角函数 >

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

解答

6 cos (x) + 6 sin (x) = 36

解答

x = 0.26179 \dots + 2 πn, x = 1.30899 \dots + 2 πn

+1

度数

x = 1 5^{\circ} + 36 0^{\circ} n, x = 7 5^{\circ} + 36 0^{\circ} n

求解步骤

6 cos (x) + 6 sin (x) = 36

两边减去

6 sin (x)

6 cos (x) = 36 - 6 sin (x)

两边进行平方

(6 cos (x))^{2} = (36 - 6 sin (x))^{2}

两边减去

(36 - 6 sin (x))^{2}

36 cos^{2} (x) - 54 + 366 sin (x) - 36 sin^{2} (x) = 0

使用三角恒等式改写

- 54 + 36 cos^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6

使用毕达哥拉斯恒等式:

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

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

= - 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 36 sin (x) 6

化简

- 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 36 sin (x) 6 : 366 sin (x) - 72 sin^{2} (x) - 18

- 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 36 sin (x) 6

= - 54 + 36 (1 - sin^{2} (x)) - 36 sin^{2} (x) + 366 sin (x)

乘开

36 (1 - sin^{2} (x)) : 36 - 36 sin^{2} (x)

36 (1 - sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 36, b = 1, c = sin^{2} (x)

= 36 \cdot 1 - 36 sin^{2} (x)

数字相乘:

36 \cdot 1 = 36

= 36 - 36 sin^{2} (x)

= - 54 + 36 - 36 sin^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6

化简

- 54 + 36 - 36 sin^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6 : 366 sin (x) - 72 sin^{2} (x) - 18

- 54 + 36 - 36 sin^{2} (x) - 36 sin^{2} (x) + 36 sin (x) 6

同类项相加:

- 36 sin^{2} (x) - 36 sin^{2} (x) = - 72 sin^{2} (x)

= - 54 + 36 - 72 sin^{2} (x) + 366 sin (x)

数字相加/相减:

- 54 + 36 = - 18

= 366 sin (x) - 72 sin^{2} (x) - 18

= 366 sin (x) - 72 sin^{2} (x) - 18

= 366 sin (x) - 72 sin^{2} (x) - 18

- 18 - 72 sin^{2} (x) + 36 sin (x) 6 = 0

用替代法求解

- 18 - 72 sin^{2} (x) + 36 sin (x) 6 = 0

令

: sin (x) = u

- 18 - 72 u^{2} + 36 u 6 = 0

- 18 - 72 u^{2} + 36 u 6 = 0 : u = \frac{6 - 2}{4}, u = \frac{6 + 2}{4}

- 18 - 72 u^{2} + 36 u 6 = 0

改写成标准形式

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

- 72 u^{2} + 366 u - 18 = 0

使用求根公式求解

- 72 u^{2} + 366 u - 18 = 0

二次方程求根公式:

若

a = - 72, b = 366, c = - 18

u_{1, 2} = \frac{- 36 6 \pm ( 36 6 ) ^{2} - 4 ( - 72 ) ( - 18 )}{2 ( - 72 )}

u_{1, 2} = \frac{- 36 6 \pm ( 36 6 ) ^{2} - 4 ( - 72 ) ( - 18 )}{2 ( - 72 )}

(366)^{2} - 4 (- 72) (- 18) = 362

(366)^{2} - 4 (- 72) (- 18)

使用法则

- (- a) = a

= (366)^{2} - 4 \cdot 72 \cdot 18

(366)^{2} = 3 6^{2} \cdot 6

(366)^{2}

使用指数法则:

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

= 3 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

= 3 6^{2} \cdot 6

4 \cdot 72 \cdot 18 = 5184

4 \cdot 72 \cdot 18

数字相乘:

4 \cdot 72 \cdot 18 = 5184

= 5184

= 3 6^{2} \cdot 6 - 5184

3 6^{2} \cdot 6 = 7776

3 6^{2} \cdot 6

3 6^{2} = 1296

= 1296 \cdot 6

数字相乘:

1296 \cdot 6 = 7776

= 7776

= 7776 - 5184

数字相减:

7776 - 5184 = 2592

= 2592

2592

质因数分解:

2^{5} \cdot 3^{4}

2592

2592

除以

22592 = 1296 \cdot 2

= 2 \cdot 1296

1296

除以

21296 = 648 \cdot 2

= 2 \cdot 2 \cdot 648

648

除以

2648 = 324 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 324

324

除以

2324 = 162 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 162

162

除以

2162 = 81 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 81

81

除以

381 = 27 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 27

27

除以

327 = 9 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 9

9

除以

39 = 3 \cdot 3

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

= 2^{5} \cdot 3^{4}

= 2^{5} \cdot 3^{4}

使用指数法则:

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

= 2^{4} \cdot 3^{4} \cdot 2

使用根式运算法则:

n ab = n a n b

= 2 2^{4} 3^{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} 2 3^{4}

使用根式运算法则:

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

3^{4} = 3^{\frac{4}{2}} = 3^{2}

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

整理后得

= 362

u_{1, 2} = \frac{- 36 6 \pm 36 2}{2 ( - 72 )}

将解分隔开

u_{1} = \frac{- 36 6 + 36 2}{2 ( - 72 )}, u_{2} = \frac{- 36 6 - 36 2}{2 ( - 72 )}

u = \frac{- 36 6 + 36 2}{2 ( - 72 )} : \frac{6 - 2}{4}

\frac{- 36 6 + 36 2}{2 ( - 72 )}

去除括号:

(- a) = - a

= \frac{- 36 6 + 36 2}{- 2 \cdot 72}

数字相乘:

2 \cdot 72 = 144

= \frac{- 36 6 + 36 2}{- 144}

使用分式法则:

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

- 366 + 362 = - (366 - 362)

= \frac{36 6 - 36 2}{144}

因式分解出通项

36

= \frac{36 ( 6 - 2 )}{144}

约分:

36

= \frac{6 - 2}{4}

u = \frac{- 36 6 - 36 2}{2 ( - 72 )} : \frac{6 + 2}{4}

\frac{- 36 6 - 36 2}{2 ( - 72 )}

去除括号:

(- a) = - a

= \frac{- 36 6 - 36 2}{- 2 \cdot 72}

数字相乘:

2 \cdot 72 = 144

= \frac{- 36 6 - 36 2}{- 144}

使用分式法则:

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

- 366 - 362 = - (366 + 362)

= \frac{36 6 + 36 2}{144}

因式分解出通项

36

= \frac{36 ( 6 + 2 )}{144}

约分:

36

= \frac{6 + 2}{4}

二次方程组的解是:

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

u = sin (x)

代回

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

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

sin (x) = \frac{6 - 2}{4} : x = arcsin (\frac{6 - 2}{4}) + 2 πn, x = π - arcsin (\frac{6 - 2}{4}) + 2 πn

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

使用反三角函数性质

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

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

的通解

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

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

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

sin (x) = \frac{6 + 2}{4} : x = arcsin (\frac{6 + 2}{4}) + 2 πn, x = π - arcsin (\frac{6 + 2}{4}) + 2 πn

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

使用反三角函数性质

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

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

的通解

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

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

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

合并所有解

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

将解代入原方程进行验证

将它们代入

6 cos (x) + 6 sin (x) = 36

检验解是否符合

去除与方程不符的解。

检验

arcsin (\frac{6 - 2}{4}) + 2 πn

的解:

真

arcsin (\frac{6 - 2}{4}) + 2 πn

代入

n = 1

arcsin (\frac{6 - 2}{4}) + 2 π 1

对于

6 cos (x) + 6 sin (x) = 36

代入

x = arcsin (\frac{6 - 2}{4}) + 2 π 1

6 cos (arcsin (\frac{6 - 2}{4}) + 2 π 1) + 6 sin (arcsin (\frac{6 - 2}{4}) + 2 π 1) = 36

整理后得

7.34846 \dots = 7.34846 \dots

\Rightarrow 真

检验

π - arcsin (\frac{6 - 2}{4}) + 2 πn

的解:

假

π - arcsin (\frac{6 - 2}{4}) + 2 πn

代入

n = 1

π - arcsin (\frac{6 - 2}{4}) + 2 π 1

对于

6 cos (x) + 6 sin (x) = 36

代入

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

6 cos (π - arcsin (\frac{6 - 2}{4}) + 2 π 1) + 6 sin (π - arcsin (\frac{6 - 2}{4}) + 2 π 1) = 36

整理后得

- 4.24264 \dots = 7.34846 \dots

\Rightarrow 假

检验

arcsin (\frac{6 + 2}{4}) + 2 πn

的解:

真

arcsin (\frac{6 + 2}{4}) + 2 πn

代入

n = 1

arcsin (\frac{6 + 2}{4}) + 2 π 1

对于

6 cos (x) + 6 sin (x) = 36

代入

x = arcsin (\frac{6 + 2}{4}) + 2 π 1

6 cos (arcsin (\frac{6 + 2}{4}) + 2 π 1) + 6 sin (arcsin (\frac{6 + 2}{4}) + 2 π 1) = 36

整理后得

7.34846 \dots = 7.34846 \dots

\Rightarrow 真

检验

π - arcsin (\frac{6 + 2}{4}) + 2 πn

的解:

假

π - arcsin (\frac{6 + 2}{4}) + 2 πn

代入

n = 1

π - arcsin (\frac{6 + 2}{4}) + 2 π 1

对于

6 cos (x) + 6 sin (x) = 36

代入

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

6 cos (π - arcsin (\frac{6 + 2}{4}) + 2 π 1) + 6 sin (π - arcsin (\frac{6 + 2}{4}) + 2 π 1) = 36

整理后得

4.24264 \dots = 7.34846 \dots

\Rightarrow 假

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

以小数形式表示解

x = 0.26179 \dots + 2 πn, x = 1.30899 \dots + 2 πn

作图

流行的例子

3 sin (x) + 1 = 0

(tan(x)+1)(cos(x)-1)=0

(tan (x) + 1) (cos (x) - 1) = 0

cot(θ)=-1/(sqrt(3))

cot (θ) = - \frac{1}{3}

3sqrt(2)cos(θ)+3=6

32 cos (θ) + 3 = 6

- 8 cos (θ) - 4 = - 4