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受欢迎的三角函数 >

2sin(x)-(2+sqrt(3))=-sqrt(3)csc(x)

解答

2 sin (x) - (2 + 3) = - 3 csc (x)

解答

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{π}{2} + 2 πn

+1

度数

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

2 sin (x) - (2 + 3) = - 3 csc (x)

两边减去

- 3 csc (x)

2 sin (x) - 2 - 3 + 3 csc (x) = 0

使用三角恒等式改写

- 2 - 3 + 2 sin (x) + csc (x) 3

使用基本三角恒等式:

sin (x) = \frac{1}{csc ( x )}

= - 2 - 3 + 2 \cdot \frac{1}{csc ( x )} + csc (x) 3

2 \cdot \frac{1}{csc ( x )} = \frac{2}{csc ( x )}

2 \cdot \frac{1}{csc ( x )}

分式相乘:

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

= \frac{1 \cdot 2}{csc ( x )}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{csc ( x )}

= - 2 - 3 + \frac{2}{csc ( x )} + 3 csc (x)

- 2 + \frac{2}{csc ( x )} - 3 + csc (x) 3 = 0

用替代法求解

- 2 + \frac{2}{csc ( x )} - 3 + csc (x) 3 = 0

令

: csc (x) = u

- 2 + \frac{2}{u} - 3 + u 3 = 0

- 2 + \frac{2}{u} - 3 + u 3 = 0 : u = \frac{2 3}{3}, u = 1

- 2 + \frac{2}{u} - 3 + u 3 = 0

在两边乘以

u

- 2 + \frac{2}{u} - 3 + u 3 = 0

在两边乘以

u

- 2 u + \frac{2}{u} u - 3 u + u 3 u = 0 \cdot u

化简

- 2 u + \frac{2}{u} u - 3 u + u 3 u = 0 \cdot u

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

化简

u 3 u : 3 u^{2}

u 3 u

使用指数法则:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

数字相加:

1 + 1 = 2

= 3 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- 2 u + 2 - 3 u + 3 u^{2} = 0

- 2 u + 2 - 3 u + 3 u^{2} = 0

- 2 u + 2 - 3 u + 3 u^{2} = 0

解

- 2 u + 2 - 3 u + 3 u^{2} = 0 : u = \frac{2 3}{3}, u = 1

- 2 u + 2 - 3 u + 3 u^{2} = 0

改写成标准形式

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

3 u^{2} - (2 + 3) u + 2 = 0

使用求根公式求解

3 u^{2} - (2 + 3) u + 2 = 0

二次方程求根公式:

若

a = 3, b = - 2 - 3, c = 2

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

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

(- 2 - 3)^{2} - 43 \cdot 2 = 2 - 3

(- 2 - 3)^{2} - 43 \cdot 2

数字相乘:

4 \cdot 2 = 8

= (- 2 - 3)^{2} - 83

乘开

(- 2 - 3)^{2} - 83 : 7 - 43

(- 2 - 3)^{2} - 83

(- 2 - 3)^{2} : 7 + 43

使用完全平方公式:

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

a = - 2, b = 3

= (- 2)^{2} - 2 (- 2) 3 + (3)^{2}

化简

(- 2)^{2} - 2 (- 2) 3 + (3)^{2} : 7 + 43

(- 2)^{2} - 2 (- 2) 3 + (3)^{2}

使用法则

- (- a) = a

= (- 2)^{2} + 2 \cdot 23 + (3)^{2}

(- 2)^{2} = 4

(- 2)^{2}

使用指数法则:

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

若

n

是偶数

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

= 2^{2}

2^{2} = 4

= 4

2 \cdot 23 = 43

2 \cdot 23

数字相乘:

2 \cdot 2 = 4

= 43

(3)^{2} = 3

(3)^{2}

使用根式运算法则:

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

= (3^{\frac{1}{2}})^{2}

使用指数法则:

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

= 3^{\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

= 3

= 4 + 43 + 3

数字相加:

4 + 3 = 7

= 7 + 43

= 7 + 43

= 7 + 43 - 83

同类项相加:

43 - 83 = - 43

= 7 - 43

= 7 - 43

= 3 - 43 + 4

= (3)^{2} - 43 + (4)^{2}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= (3)^{2} - 43 + 2^{2}

23 \cdot 2 = 43

23 \cdot 2

数字相乘:

2 \cdot 2 = 4

= 43

= (3)^{2} - 23 \cdot 2 + 2^{2}

使用完全平方公式:

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

(3)^{2} - 23 \cdot 2 + 2^{2} = (3 - 2)^{2}

= (3 - 2)^{2}

使用指数法则:

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

若

n

是偶数

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

= (2 - 3)^{2}

使用根式运算法则:

n a^{n} = a

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

= 2 - 3

u_{1, 2} = \frac{- ( - 2 - 3 ) \pm ( 2 - 3 )}{2 3}

将解分隔开

u_{1} = \frac{- ( - 2 - 3 ) + 2 - 3}{2 3}, u_{2} = \frac{- ( - 2 - 3 ) - ( 2 - 3 )}{2 3}

u = \frac{- ( - 2 - 3 ) + 2 - 3}{2 3} : \frac{2 3}{3}

\frac{- ( - 2 - 3 ) + 2 - 3}{2 3}

乘开

- (- 2 - 3) + 2 - 3 : 4

- (- 2 - 3) + 2 - 3

- (- 2 - 3) : 2 + 3

- (- 2 - 3)

打开括号

= - (- 2) - (- 3)

使用加减运算法则

- (- a) = a

= 2 + 3

= 2 + 3 + 2 - 3

化简

2 + 3 + 2 - 3 : 4

2 + 3 + 2 - 3

同类项相加:

3 - 3 = 0

= 2 + 2

数字相加:

2 + 2 = 4

= 4

= 4

= \frac{4}{2 3}

数字相除:

\frac{4}{2} = 2

= \frac{2}{3}

\frac{2}{3}

有理化:

\frac{2 3}{3}

\frac{2}{3}

乘以共轭根式

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

u = \frac{- ( - 2 - 3 ) - ( 2 - 3 )}{2 3} : 1

\frac{- ( - 2 - 3 ) - ( 2 - 3 )}{2 3}

乘开

- (- 2 - 3) - (2 - 3) : 23

- (- 2 - 3) - (2 - 3)

- (- 2 - 3) : 2 + 3

- (- 2 - 3)

打开括号

= - (- 2) - (- 3)

使用加减运算法则

- (- a) = a

= 2 + 3

= 2 + 3 - (2 - 3)

- (2 - 3) : - 2 + 3

- (2 - 3)

打开括号

= - (2) - (- 3)

使用加减运算法则

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

= - 2 + 3

= 2 + 3 - 2 + 3

化简

2 + 3 - 2 + 3 : 23

2 + 3 - 2 + 3

同类项相加:

3 + 3 = 23

= 2 + 23 - 2

2 - 2 = 0

= 23

= 23

= \frac{2 3}{2 3}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

u = \frac{2 3}{3}, u = 1

u = \frac{2 3}{3}, u = 1

验证解

找到无定义的点（奇点）:

u = 0

取

- 2 + \frac{2}{u} - 3 + u 3

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{2 3}{3}, u = 1

u = csc (x)

代回

csc (x) = \frac{2 3}{3}, csc (x) = 1

csc (x) = \frac{2 3}{3}, csc (x) = 1

csc (x) = \frac{2 3}{3} : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

csc (x) = \frac{2 3}{3}

csc (x) = \frac{2 3}{3}

的通解

csc (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

csc (x) = 1

的通解

csc (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

合并所有解

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{π}{2} + 2 πn

作图

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