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受欢迎的三角函数 >

cot^2(x)csc^2(x)-cot^2(x)=9

解答

cot^{2} (x) csc^{2} (x) - cot^{2} (x) = 9

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

cot^{2} (x) csc^{2} (x) - cot^{2} (x) = 9

两边减去

9

cot^{2} (x) csc^{2} (x) - cot^{2} (x) - 9 = 0

使用三角恒等式改写

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

使用毕达哥拉斯恒等式:

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

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

= - 9 - (csc^{2} (x) - 1) + (csc^{2} (x) - 1) csc^{2} (x)

化简

- 9 - (csc^{2} (x) - 1) + (csc^{2} (x) - 1) csc^{2} (x) : csc^{4} (x) - 2 csc^{2} (x) - 8

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

= - 9 - (csc^{2} (x) - 1) + csc^{2} (x) (csc^{2} (x) - 1)

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

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

打开括号

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

使用加减运算法则

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

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

= - 9 - csc^{2} (x) + 1 + (csc^{2} (x) - 1) csc^{2} (x)

乘开

csc^{2} (x) (csc^{2} (x) - 1) : csc^{4} (x) - csc^{2} (x)

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

使用分配律:

a (b - c) = ab - a c

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

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

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

化简

csc^{2} (x) csc^{2} (x) - 1 \cdot csc^{2} (x) : csc^{4} (x) - csc^{2} (x)

csc^{2} (x) csc^{2} (x) - 1 \cdot csc^{2} (x)

csc^{2} (x) csc^{2} (x) = csc^{4} (x)

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

使用指数法则:

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

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

= csc^{2 + 2} (x)

数字相加:

2 + 2 = 4

= csc^{4} (x)

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

1 \cdot csc^{2} (x)

乘以:

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

= csc^{2} (x)

= csc^{4} (x) - csc^{2} (x)

= csc^{4} (x) - csc^{2} (x)

= - 9 - csc^{2} (x) + 1 + csc^{4} (x) - csc^{2} (x)

化简

- 9 - csc^{2} (x) + 1 + csc^{4} (x) - csc^{2} (x) : csc^{4} (x) - 2 csc^{2} (x) - 8

- 9 - csc^{2} (x) + 1 + csc^{4} (x) - csc^{2} (x)

对同类项分组

= - csc^{2} (x) + csc^{4} (x) - csc^{2} (x) - 9 + 1

同类项相加:

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

= - 2 csc^{2} (x) + csc^{4} (x) - 9 + 1

数字相加/相减:

- 9 + 1 = - 8

= csc^{4} (x) - 2 csc^{2} (x) - 8

= csc^{4} (x) - 2 csc^{2} (x) - 8

= csc^{4} (x) - 2 csc^{2} (x) - 8

- 8 + csc^{4} (x) - 2 csc^{2} (x) = 0

用替代法求解

- 8 + csc^{4} (x) - 2 csc^{2} (x) = 0

令

: csc (x) = u

- 8 + u^{4} - 2 u^{2} = 0

- 8 + u^{4} - 2 u^{2} = 0 : u = 2, u = - 2, u = 2 i, u = - 2 i

- 8 + u^{4} - 2 u^{2} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - 2 u^{2} - 8 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} - 2 v - 8 = 0

解

v^{2} - 2 v - 8 = 0 : v = 4, v = - 2

v^{2} - 2 v - 8 = 0

使用求根公式求解

v^{2} - 2 v - 8 = 0

二次方程求根公式:

若

a = 1, b = - 2, c = - 8

v_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot ( - 8 )}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 1 \cdot ( - 8 )}{2 \cdot 1}

(- 2)^{2} - 4 \cdot 1 \cdot (- 8) = 6

(- 2)^{2} - 4 \cdot 1 \cdot (- 8)

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 1 \cdot 8

使用指数法则:

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

若

n

是偶数

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

= 2^{2} + 4 \cdot 1 \cdot 8

数字相乘:

4 \cdot 1 \cdot 8 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数字相加:

4 + 32 = 36

= 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

v_{1, 2} = \frac{- ( - 2 ) \pm 6}{2 \cdot 1}

将解分隔开

v_{1} = \frac{- ( - 2 ) + 6}{2 \cdot 1}, v_{2} = \frac{- ( - 2 ) - 6}{2 \cdot 1}

v = \frac{- ( - 2 ) + 6}{2 \cdot 1} : 4

\frac{- ( - 2 ) + 6}{2 \cdot 1}

使用法则

- (- a) = a

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

数字相加:

2 + 6 = 8

= \frac{8}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8}{2}

数字相除:

\frac{8}{2} = 4

= 4

v = \frac{- ( - 2 ) - 6}{2 \cdot 1} : - 2

\frac{- ( - 2 ) - 6}{2 \cdot 1}

使用法则

- (- a) = a

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

数字相减:

2 - 6 = - 4

= \frac{- 4}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 4}{2}

使用分式法则:

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

= - \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= - 2

二次方程组的解是:

v = 4, v = - 2

v = 4, v = - 2

代回

v = u^{2}

，求解

u

解

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

对于

x^{2} = f (a)

解为

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

u = 4, u = - 4

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

- 4 = - 2

- 4

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

= - 2

u = 2, u = - 2

解

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

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

解为

u = 2, u = - 2, u = 2 i, u = - 2 i

u = csc (x)

代回

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

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

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

csc (x) = 2

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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = - 2

csc (x) = - 2

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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) = 2 i :

无解

csc (x) = 2 i

无解

csc (x) = - 2 i :

无解

csc (x) = - 2 i

无解

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

作图

流行的例子

2sin(θ)+sin(2θ)=0

2 sin (θ) + sin (2 θ) = 0

pi/4 =arctan(x)

\frac{π}{4} = arctan (x)

2cos(x)=cos(2x)

2 cos (x) = cos (2 x)

3cos^2(θ)+cos(θ)=0

3 cos^{2} (θ) + cos (θ) = 0

cos^{2} (θ) = 0.75