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受欢迎的三角函数 >

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

解答

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

解答

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

2

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

使用三角恒等式改写

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

2 \cdot 1 = 2

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

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

化简

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

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

对同类项分组

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

数字相减:

- 2 - 2 = - 4

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

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

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

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

用替代法求解

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

令

: csc (x) = u

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

- 4 - 2 u + 2 u^{2} = 0 : u = 2, u = - 1

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 2, b = - 2, c = - 4

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

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

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

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 2 \cdot 4 = 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

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

将解分隔开

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

u = \frac{- ( - 2 ) + 6}{2 \cdot 2} : 2

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

使用法则

- (- a) = a

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

数字相加:

2 + 6 = 8

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

数字相乘:

2 \cdot 2 = 4

= \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= 2

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

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

使用法则

- (- a) = a

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

数字相减:

2 - 6 = - 4

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{4}

使用分式法则:

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

= - \frac{4}{4}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = 2, u = - 1

u = csc (x)

代回

csc (x) = 2, csc (x) = - 1

csc (x) = 2, csc (x) = - 1

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) = - 1 : x = \frac{3 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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

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