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受欢迎的三角函数 >

cot^2(θ)= 3/2 csc(θ)

解答

cot^{2} (θ) = \frac{3}{2} csc (θ)

解答

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

+1

度数

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

求解步骤

cot^{2} (θ) = \frac{3}{2} csc (θ)

两边减去

\frac{3}{2} csc (θ)

cot^{2} (θ) - \frac{3}{2} csc (θ) = 0

化简

cot^{2} (θ) - \frac{3}{2} csc (θ) : \frac{2 cot ^{2} ( θ ) - 3 csc ( θ )}{2}

cot^{2} (θ) - \frac{3}{2} csc (θ)

乘

\frac{3}{2} csc (θ) : \frac{3 csc ( θ )}{2}

\frac{3}{2} csc (θ)

分式相乘:

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

= \frac{3 csc ( θ )}{2}

= cot^{2} (θ) - \frac{3 csc ( θ )}{2}

将项转换为分式:

cot^{2} (θ) = \frac{cot ^{2} ( θ ) 2}{2}

= \frac{cot ^{2} ( θ ) \cdot 2}{2} - \frac{3 csc ( θ )}{2}

因为分母相等，所以合并分式:

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

= \frac{cot ^{2} ( θ ) \cdot 2 - 3 csc ( θ )}{2}

\frac{2 cot ^{2} ( θ ) - 3 csc ( θ )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 cot^{2} (θ) - 3 csc (θ) = 0

使用三角恒等式改写

2 cot^{2} (θ) - 3 csc (θ)

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 (csc^{2} (θ) - 1) - 3 csc (θ)

(- 1 + csc^{2} (θ)) \cdot 2 - 3 csc (θ) = 0

用替代法求解

(- 1 + csc^{2} (θ)) \cdot 2 - 3 csc (θ) = 0

令

: csc (θ) = u

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

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

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

展开

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

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

= 2 (- 1 + u^{2}) - 3 u

乘开

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

2 (- 1 + u^{2})

使用分配律:

a (b + c) = ab + a c

a = 2, b = - 1, c = u^{2}

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

使用加减运算法则

+ (- a) = - a

= - 2 \cdot 1 + 2 u^{2}

数字相乘:

2 \cdot 1 = 2

= - 2 + 2 u^{2}

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

(- 3)^{2} - 4 \cdot 2 (- 2) = 5

(- 3)^{2} - 4 \cdot 2 (- 2)

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 2 \cdot 2

使用指数法则:

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

若

n

是偶数

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

= 3^{2} + 4 \cdot 2 \cdot 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 3^{2} + 16

3^{2} = 9

= 9 + 16

数字相加:

9 + 16 = 25

= 25

因式分解数字:

25 = 5^{2}

= 5^{2}

使用根式运算法则:

n a^{n} = a

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 5}{2 \cdot 2}, u_{2} = \frac{- ( - 3 ) - 5}{2 \cdot 2}

u = \frac{- ( - 3 ) + 5}{2 \cdot 2} : 2

\frac{- ( - 3 ) + 5}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 + 5}{2 \cdot 2}

数字相加:

3 + 5 = 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{- ( - 3 ) - 5}{2 \cdot 2} : - \frac{1}{2}

\frac{- ( - 3 ) - 5}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{3 - 5}{2 \cdot 2}

数字相减:

3 - 5 = - 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

二次方程组的解是:

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

u = csc (θ)

代回

csc (θ) = 2, csc (θ) = - \frac{1}{2}

csc (θ) = 2, csc (θ) = - \frac{1}{2}

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

csc (θ) = 2

csc (θ) = 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

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

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

csc (θ) = - \frac{1}{2} :

无解

csc (θ) = - \frac{1}{2}

csc (x) \leq - 1 or csc (x) \geq 1

无解

合并所有解

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

作图

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