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cot^2(x)csc^2(x)+2csc^2(x)-cot^2(x)=2

解答

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) = 2

解答

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

+1

度数

x = 27 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

求解步骤

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) = 2

两边减去

2

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) - 2 = 0

分解

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) - 2 : (cot^{2} (x) + 2) (csc (x) + 1) (csc (x) - 1)

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) - 2

= (cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x)) + (- cot^{2} (x) - 2)

从

- cot^{2} (x) - 2

分解出因式

- 1

:

- (cot^{2} (x) + 2)

- cot^{2} (x) - 2

因式分解出通项

- 1

= - (cot^{2} (x) + 2)

从

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x)

分解出因式

csc^{2} (x)

:

csc^{2} (x) (cot^{2} (x) + 2)

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x)

因式分解出通项

csc^{2} (x)

= csc^{2} (x) (cot^{2} (x) + 2)

= - (cot^{2} (x) + 2) + csc^{2} (x) (cot^{2} (x) + 2)

因式分解出通项

cot^{2} (x) + 2

= (cot^{2} (x) + 2) (csc^{2} (x) - 1)

分解

csc^{2} (x) - 1 : (csc (x) + 1) (csc (x) - 1)

csc^{2} (x) - 1

将

1

改写为

1^{2}

= csc^{2} (x) - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

csc^{2} (x) - 1^{2} = (csc (x) + 1) (csc (x) - 1)

= (csc (x) + 1) (csc (x) - 1)

= (cot^{2} (x) + 2) (csc (x) + 1) (csc (x) - 1)

(cot^{2} (x) + 2) (csc (x) + 1) (csc (x) - 1) = 0

分别求解每个部分

cot^{2} (x) + 2 = 0 or csc (x) + 1 = 0 or csc (x) - 1 = 0

cot^{2} (x) + 2 = 0 :

无解

cot^{2} (x) + 2 = 0

用替代法求解

cot^{2} (x) + 2 = 0

令

: cot (x) = u

u^{2} + 2 = 0

u^{2} + 2 = 0 : u = 2 i, u = - 2 i

u^{2} + 2 = 0

将

2

到右边

u^{2} + 2 = 0

两边减去

2

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

化简

u^{2} = - 2

u^{2} = - 2

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = - 2, u = - - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

化简

- - 2 : - 2 i

- - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

u = cot (x)

代回

cot (x) = 2 i, cot (x) = - 2 i

cot (x) = 2 i, cot (x) = - 2 i

cot (x) = 2 i :

无解

cot (x) = 2 i

无解

cot (x) = - 2 i :

无解

cot (x) = - 2 i

无解

合并所有解

无解

csc (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

csc (x) + 1 = 0

将

1

到右边

csc (x) + 1 = 0

两边减去

1

csc (x) + 1 - 1 = 0 - 1

化简

csc (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

csc (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

csc (x) - 1 = 0

将

1

到右边

csc (x) - 1 = 0

两边加上

1

csc (x) - 1 + 1 = 0 + 1

化简

csc (x) = 1

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

流行的例子

cos(t)= 3/5 ,0<t< pi/2

cos (t) = \frac{3}{5}, 0 < t < \frac{π}{2}

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

solvefor y,-ln(cos(x))=-2cos(x)*sin(y)求解

y, - ln (cos (x)) = - 2 cos (x) \cdot sin (y)

2sin(a)+csc(a)=3

2 sin (a) + csc (a) = 3

cos (x) = \frac{- 2}{7}