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受欢迎的三角函数 >

2-sqrt(-1-6sin(x))=sqrt(-4sin(x))

解答

2 - - 1 - 6 sin (x) = - 4 sin (x)

解答

x = - 0.30674 \dots + 2 πn, x = π + 0.30674 \dots + 2 πn

+1

度数

x = - 17.57542 \dots^{\circ} + 36 0^{\circ} n, x = 197.57542 \dots^{\circ} + 36 0^{\circ} n

求解步骤

2 - - 1 - 6 sin (x) = - 4 sin (x)

用替代法求解

2 - - 1 - 6 sin (x) = - 4 sin (x)

令

: sin (x) = u

2 - - 1 - 6 u = - 4 u

2 - - 1 - 6 u = - 4 u : u = \frac{- 21 + 4 26}{2}

2 - - 1 - 6 u = - 4 u

去除平方根

2 - - 1 - 6 u = - 4 u

两边进行平方:

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

2 - - 1 - 6 u = - 4 u

(2 - - 1 - 6 u)^{2} = (- 4 u)^{2}

展开

(2 - - 1 - 6 u)^{2} : - 6 u + 3 - 4 - 1 - 6 u

(2 - - 1 - 6 u)^{2}

使用完全平方公式:

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

a = 2, b = - 1 - 6 u

= 2^{2} - 2 \cdot 2 - 1 - 6 u + (- 1 - 6 u)^{2}

化简

2^{2} - 2 \cdot 2 - 1 - 6 u + (- 1 - 6 u)^{2} : 4 - 4 - 1 - 6 u + - 1 - 6 u

2^{2} - 2 \cdot 2 - 1 - 6 u + (- 1 - 6 u)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 - 1 - 6 u = 4 - 1 - 6 u

2 \cdot 2 - 1 - 6 u

数字相乘:

2 \cdot 2 = 4

= 4 - 1 - 6 u

(- 1 - 6 u)^{2} = - 1 - 6 u

(- 1 - 6 u)^{2}

使用根式运算法则:

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

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

使用指数法则:

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

= (- 1 - 6 u)^{\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

= - 1 - 6 u

= 4 - 4 - 1 - 6 u - 1 - 6 u

= 4 - 4 - 1 - 6 u - 1 - 6 u

整理后得

= - 6 u + 3 - 4 - 1 - 6 u

展开

(- 4 u)^{2} : - 4 u

(- 4 u)^{2}

使用根式运算法则:

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

= ((- 4 u)^{\frac{1}{2}})^{2}

使用指数法则:

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

= (- 4 u)^{\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

= - 4 u

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

两边加上

6 u

- 6 u + 3 - 4 - 1 - 6 u + 6 u = - 4 u + 6 u

化简

- 4 - 1 - 6 u + 3 = 2 u

两边减去

3

- 4 - 1 - 6 u + 3 - 3 = 2 u - 3

化简

- 4 - 1 - 6 u = 2 u - 3

两边进行平方:

- 16 - 96 u = 4 u^{2} - 12 u + 9

- 6 u + 3 - 4 - 1 - 6 u = - 4 u

(- 4 - 1 - 6 u)^{2} = (2 u - 3)^{2}

展开

(- 4 - 1 - 6 u)^{2} : - 16 - 96 u

(- 4 - 1 - 6 u)^{2}

使用指数法则:

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

若

n

是偶数

(- 4 - 1 - 6 u)^{2} = (4 - 1 - 6 u)^{2}

= (4 - 1 - 6 u)^{2}

使用指数法则:

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

= 4^{2} (- 1 - 6 u)^{2}

(- 1 - 6 u)^{2} : - 1 - 6 u

使用根式运算法则:

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

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

使用指数法则:

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

= (- 1 - 6 u)^{\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

= - 1 - 6 u

= 4^{2} (- 1 - 6 u)

4^{2} = 16

= 16 (- 1 - 6 u)

展开

16 (- 1 - 6 u) : - 16 - 96 u

16 (- 1 - 6 u)

使用分配律:

a (b - c) = ab - a c

a = 16, b = - 1, c = 6 u

= 16 (- 1) - 16 \cdot 6 u

使用加减运算法则

+ (- a) = - a

= - 16 \cdot 1 - 16 \cdot 6 u

化简

- 16 \cdot 1 - 16 \cdot 6 u : - 16 - 96 u

- 16 \cdot 1 - 16 \cdot 6 u

数字相乘:

16 \cdot 1 = 16

= - 16 - 16 \cdot 6 u

数字相乘:

16 \cdot 6 = 96

= - 16 - 96 u

= - 16 - 96 u

= - 16 - 96 u

展开

(2 u - 3)^{2} : 4 u^{2} - 12 u + 9

(2 u - 3)^{2}

使用完全平方公式:

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

a = 2 u, b = 3

= (2 u)^{2} - 2 \cdot 2 u \cdot 3 + 3^{2}

化简

(2 u)^{2} - 2 \cdot 2 u \cdot 3 + 3^{2} : 4 u^{2} - 12 u + 9

(2 u)^{2} - 2 \cdot 2 u \cdot 3 + 3^{2}

(2 u)^{2} = 4 u^{2}

(2 u)^{2}

使用指数法则:

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

= 2^{2} u^{2}

2^{2} = 4

= 4 u^{2}

2 \cdot 2 u \cdot 3 = 12 u

2 \cdot 2 u \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 = 12

= 12 u

3^{2} = 9

3^{2}

3^{2} = 9

= 9

= 4 u^{2} - 12 u + 9

= 4 u^{2} - 12 u + 9

- 16 - 96 u = 4 u^{2} - 12 u + 9

- 16 - 96 u = 4 u^{2} - 12 u + 9

- 16 - 96 u = 4 u^{2} - 12 u + 9

解

- 16 - 96 u = 4 u^{2} - 12 u + 9 : u = \frac{- 21 + 4 26}{2}, u = - \frac{21 + 4 26}{2}

- 16 - 96 u = 4 u^{2} - 12 u + 9

交换两边

4 u^{2} - 12 u + 9 = - 16 - 96 u

将

96 u

para o lado esquerdo

4 u^{2} - 12 u + 9 = - 16 - 96 u

两边加上

96 u

4 u^{2} - 12 u + 9 + 96 u = - 16 - 96 u + 96 u

化简

4 u^{2} + 84 u + 9 = - 16

4 u^{2} + 84 u + 9 = - 16

将

16

para o lado esquerdo

4 u^{2} + 84 u + 9 = - 16

两边加上

16

4 u^{2} + 84 u + 9 + 16 = - 16 + 16

化简

4 u^{2} + 84 u + 25 = 0

4 u^{2} + 84 u + 25 = 0

使用求根公式求解

4 u^{2} + 84 u + 25 = 0

二次方程求根公式:

若

a = 4, b = 84, c = 25

u_{1, 2} = \frac{- 84 \pm 8 4 ^{2} - 4 \cdot 4 \cdot 25}{2 \cdot 4}

u_{1, 2} = \frac{- 84 \pm 8 4 ^{2} - 4 \cdot 4 \cdot 25}{2 \cdot 4}

8 4^{2} - 4 \cdot 4 \cdot 25 = 1626

8 4^{2} - 4 \cdot 4 \cdot 25

数字相乘:

4 \cdot 4 \cdot 25 = 400

= 8 4^{2} - 400

8 4^{2} = 7056

= 7056 - 400

数字相减:

7056 - 400 = 6656

= 6656

6656

质因数分解:

2^{9} \cdot 13

6656

6656

除以

26656 = 3328 \cdot 2

= 2 \cdot 3328

3328

除以

23328 = 1664 \cdot 2

= 2 \cdot 2 \cdot 1664

1664

除以

21664 = 832 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 832

832

除以

2832 = 416 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 416

416

除以

2416 = 208 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 208

208

除以

2208 = 104 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 104

104

除以

2104 = 52 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 52

52

除以

252 = 26 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 26

26

除以

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 13

2, 13

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 2 \cdot 2 \cdot 2 \cdot 2 \cdot 13

= 2^{9} \cdot 13

= 2^{9} \cdot 13

使用指数法则:

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

= 2^{8} \cdot 2 \cdot 13

使用根式运算法则:

n ab = n a n b

= 2^{8} 2 \cdot 13

使用根式运算法则:

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

2^{8} = 2^{\frac{8}{2}} = 2^{4}

= 2^{4} 2 \cdot 13

整理后得

= 1626

u_{1, 2} = \frac{- 84 \pm 16 26}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 84 + 16 26}{2 \cdot 4}, u_{2} = \frac{- 84 - 16 26}{2 \cdot 4}

u = \frac{- 84 + 16 26}{2 \cdot 4} : \frac{- 21 + 4 26}{2}

\frac{- 84 + 16 26}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 84 + 16 26}{8}

分解

- 84 + 1626 : 4 (- 21 + 426)

- 84 + 1626

改写为

= - 4 \cdot 21 + 4 \cdot 426

因式分解出通项

4

= 4 (- 21 + 426)

= \frac{4 ( - 21 + 4 26 )}{8}

约分:

4

= \frac{- 21 + 4 26}{2}

u = \frac{- 84 - 16 26}{2 \cdot 4} : - \frac{21 + 4 26}{2}

\frac{- 84 - 16 26}{2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 84 - 16 26}{8}

分解

- 84 - 1626 : - 4 (21 + 426)

- 84 - 1626

改写为

= - 4 \cdot 21 - 4 \cdot 426

因式分解出通项

4

= - 4 (21 + 426)

= - \frac{4 ( 21 + 4 26 )}{8}

约分:

4

= - \frac{21 + 4 26}{2}

二次方程组的解是:

u = \frac{- 21 + 4 26}{2}, u = - \frac{21 + 4 26}{2}

u = \frac{- 21 + 4 26}{2}, u = - \frac{21 + 4 26}{2}

验证解:

u = \frac{- 21 + 4 26}{2}

真

, u = - \frac{21 + 4 26}{2}

假

将它们代入

2 - - 1 - 6 u = - 4 u

检验解是否符合

去除与方程不符的解。

代入

u = \frac{- 21 + 4 26}{2} :

真

2 - - 1 - 6 (\frac{- 21 + 4 26}{2}) = - 4 (\frac{- 21 + 4 26}{2})

2 - - 1 - 6 (\frac{- 21 + 4 26}{2}) = 26 - 4

2 - - 1 - 6 (\frac{- 21 + 4 26}{2})

去除括号:

(a) = a

= 2 - - 1 - 6 \cdot \frac{- 21 + 4 26}{2}

- 1 - 6 \cdot \frac{- 21 + 4 26}{2} = 6 - 26

- 1 - 6 \cdot \frac{- 21 + 4 26}{2}

6 \cdot \frac{- 21 + 4 26}{2} = 3 (426 - 21)

6 \cdot \frac{- 21 + 4 26}{2}

分式相乘:

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

= \frac{( - 21 + 4 26 ) \cdot 6}{2}

数字相除:

\frac{6}{2} = 3

= 3 (426 - 21)

= - 1 - 3 (426 - 21)

乘开

- 1 - 3 (426 - 21) : 62 - 1226

- 1 - 3 (426 - 21)

乘开

- 3 (426 - 21) : - 1226 + 63

- 3 (426 - 21)

使用分配律:

a (b - c) = ab - a c

a = - 3, b = 426, c = 21

= - 3 \cdot 426 - (- 3) \cdot 21

使用加减运算法则

- (- a) = a

= - 3 \cdot 426 + 3 \cdot 21

化简

- 3 \cdot 426 + 3 \cdot 21 : - 1226 + 63

- 3 \cdot 426 + 3 \cdot 21

数字相乘:

3 \cdot 4 = 12

= - 1226 + 3 \cdot 21

数字相乘:

3 \cdot 21 = 63

= - 1226 + 63

= - 1226 + 63

= - 1 - 1226 + 63

数字相加/相减:

- 1 + 63 = 62

= 62 - 1226

= 62 - 1226

= 26 - 1226 + 36

= (26)^{2} - 1226 + (36)^{2}

36 = 6

36

因式分解数字:

36 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a

6^{2} = 6

= 6

= (26)^{2} - 1226 + 6^{2}

226 \cdot 6 = 1226

226 \cdot 6

数字相乘:

2 \cdot 6 = 12

= 1226

= (26)^{2} - 226 \cdot 6 + 6^{2}

使用完全平方公式:

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

(26)^{2} - 226 \cdot 6 + 6^{2} = (26 - 6)^{2}

= (26 - 6)^{2}

使用指数法则:

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

若

n

是偶数

(26 - 6)^{2} = (6 - 26)^{2}

= (6 - 26)^{2}

使用根式运算法则:

n a^{n} = a

(6 - 26)^{2} = 6 - 26

= 6 - 26

= 2 - (6 - 26)

- (6 - 26) : - 6 + 26

- (6 - 26)

打开括号

= - (6) - (- 26)

使用加减运算法则

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

= - 6 + 26

= 2 - 6 + 26

数字相减:

2 - 6 = - 4

= 26 - 4

- 4 (\frac{- 21 + 4 26}{2}) = 2 - 426 + 21

- 4 (\frac{- 21 + 4 26}{2})

去除括号:

(a) = a

= - 4 \cdot \frac{- 21 + 4 26}{2}

乘

- 4 \cdot \frac{- 21 + 4 26}{2} : - 2 (426 - 21)

- 4 \cdot \frac{- 21 + 4 26}{2}

分式相乘:

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

= - \frac{( - 21 + 4 26 ) \cdot 4}{2}

数字相除:

\frac{4}{2} = 2

= - 2 (426 - 21)

= - 2 (426 - 21)

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= 2 - (426 - 21)

乘开

- (426 - 21) : - 426 + 21

- (426 - 21)

打开括号

= - (426) - (- 21)

使用加减运算法则

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

= - 426 + 21

= 2 21 - 426

26 - 4 = 2 - 426 + 21

真

代入

u = - \frac{21 + 4 26}{2} :

假

2 - - 1 - 6 (- \frac{21 + 4 26}{2}) = - 4 (- \frac{21 + 4 26}{2})

2 - - 1 - 6 (- \frac{21 + 4 26}{2}) = - 4 - 26

2 - - 1 - 6 (- \frac{21 + 4 26}{2})

使用法则

- (- a) = a

= 2 - - 1 + 6 \cdot \frac{21 + 4 26}{2}

- 1 + 6 \cdot \frac{21 + 4 26}{2} = 26 + 6

- 1 + 6 \cdot \frac{21 + 4 26}{2}

6 \cdot \frac{21 + 4 26}{2} = 3 (21 + 426)

6 \cdot \frac{21 + 4 26}{2}

分式相乘:

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

= \frac{( 21 + 4 26 ) \cdot 6}{2}

数字相除:

\frac{6}{2} = 3

= 3 (21 + 426)

= - 1 + 3 (21 + 426)

乘开

- 1 + 3 (21 + 426) : 62 + 1226

- 1 + 3 (21 + 426)

乘开

3 (21 + 426) : 63 + 1226

3 (21 + 426)

使用分配律:

a (b + c) = ab + a c

a = 3, b = 21, c = 426

= 3 \cdot 21 + 3 \cdot 426

化简

3 \cdot 21 + 3 \cdot 426 : 63 + 1226

3 \cdot 21 + 3 \cdot 426

数字相乘:

3 \cdot 21 = 63

= 63 + 3 \cdot 426

数字相乘:

3 \cdot 4 = 12

= 63 + 1226

= 63 + 1226

= - 1 + 63 + 1226

数字相加/相减:

- 1 + 63 = 62

= 62 + 1226

= 62 + 1226

= 26 + 1226 + 36

= (26)^{2} + 1226 + (36)^{2}

36 = 6

36

因式分解数字:

36 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a

6^{2} = 6

= 6

= (26)^{2} + 1226 + 6^{2}

226 \cdot 6 = 1226

226 \cdot 6

数字相乘:

2 \cdot 6 = 12

= 1226

= (26)^{2} + 226 \cdot 6 + 6^{2}

使用完全平方公式:

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

(26)^{2} + 226 \cdot 6 + 6^{2} = (26 + 6)^{2}

= (26 + 6)^{2}

使用根式运算法则:

n a^{n} = a

(26 + 6)^{2} = 26 + 6

= 26 + 6

= 2 - (6 + 26)

- (26 + 6) : - 26 - 6

- (26 + 6)

打开括号

= - (26) - (6)

使用加减运算法则

+ (- a) = - a

= - 26 - 6

= 2 - 26 - 6

数字相减:

2 - 6 = - 4

= - 4 - 26

- 4 (- \frac{21 + 4 26}{2}) = 2 21 + 426

- 4 (- \frac{21 + 4 26}{2})

使用法则

- (- a) = a

= 4 \cdot \frac{21 + 4 26}{2}

乘

4 \cdot \frac{21 + 4 26}{2} : 2 (21 + 426)

4 \cdot \frac{21 + 4 26}{2}

分式相乘:

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

= \frac{( 21 + 4 26 ) \cdot 4}{2}

数字相除:

\frac{4}{2} = 2

= 2 (21 + 426)

= 2 (21 + 426)

使用根式运算法则:

n ab = n a n b,

假定

a \geq 0, b \geq 0

= 2 21 + 426

- 4 - 26 = 2 21 + 426

假

解是

u = \frac{- 21 + 4 26}{2}

u = sin (x)

代回

sin (x) = \frac{- 21 + 4 26}{2}

sin (x) = \frac{- 21 + 4 26}{2}

sin (x) = \frac{- 21 + 4 26}{2} : x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

sin (x) = \frac{- 21 + 4 26}{2}

使用反三角函数性质

sin (x) = \frac{- 21 + 4 26}{2}

sin (x) = \frac{- 21 + 4 26}{2}

的通解

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

x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

合并所有解

x = arcsin (\frac{- 21 + 4 26}{2}) + 2 πn, x = π + arcsin (- \frac{- 21 + 4 26}{2}) + 2 πn

以小数形式表示解

x = - 0.30674 \dots + 2 πn, x = π + 0.30674 \dots + 2 πn

作图

流行的例子

sqrt(2)sin(x/2)-1=0

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cos^2(x)-sin^2(x)+sin(x)=0

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

2tan^2(t)=-3sec(t)

2 tan^{2} (t) = - 3 sec (t)

tan^{2} (x) + 3 = 0

4 cos (θ) + 1 = 0