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受欢迎的三角函数 >

1/(cot^2(x))+(sqrt(3))/(cot(x))=0

解答

\frac{1}{cot ^{2} ( x )} + \frac{3}{cot ( x )} = 0

解答

x = \frac{2 π}{3} + πn

+1

度数

x = 12 0^{\circ} + 18 0^{\circ} n

求解步骤

\frac{1}{cot ^{2} ( x )} + \frac{3}{cot ( x )} = 0

用替代法求解

\frac{1}{cot ^{2} ( x )} + \frac{3}{cot ( x )} = 0

令

: cot (x) = u

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

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

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

乘以最小公倍数

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

找到

u^{2}, u

的最小公倍数:

u^{2}

u^{2}, u

最小公倍数 (LCM)

计算出由出现在

u^{2}

或

u

中的因子组成的表达式

= u^{2}

乘以最小公倍数=

u^{2}

\frac{1}{u ^{2}} u^{2} + \frac{3}{u} u^{2} = 0 \cdot u^{2}

化简

\frac{1}{u ^{2}} u^{2} + \frac{3}{u} u^{2} = 0 \cdot u^{2}

化简

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

分式相乘:

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

= \frac{1 \cdot u ^{2}}{u ^{2}}

约分:

u^{2}

= 1

化简

\frac{3}{u} u^{2} : 3 u

\frac{3}{u} u^{2}

分式相乘:

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

= \frac{3 u ^{2}}{u}

约分:

u

= 3 u

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

1 + 3 u = 0

1 + 3 u = 0

1 + 3 u = 0

将

1

到右边

1 + 3 u = 0

两边减去

1

1 + 3 u - 1 = 0 - 1

化简

3 u = - 1

3 u = - 1

两边除以

3

3 u = - 1

两边除以

3

\frac{3 u}{3} = \frac{- 1}{3}

化简

\frac{3 u}{3} = \frac{- 1}{3}

化简

\frac{3 u}{3} : u

\frac{3 u}{3}

约分:

3

= u

化简

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

使用分式法则:

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

= - \frac{1}{3}

- \frac{1}{3}

有理化:

- \frac{3}{3}

- \frac{1}{3}

乘以共轭根式

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

验证解

找到无定义的点（奇点）:

u = 0

取

\frac{1}{u ^{2}} + \frac{3}{u}

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

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{3}{3}

u = cot (x)

代回

cot (x) = - \frac{3}{3}

cot (x) = - \frac{3}{3}

cot (x) = - \frac{3}{3} : x = \frac{2 π}{3} + πn

cot (x) = - \frac{3}{3}

cot (x) = - \frac{3}{3}

的通解

cot (x)

周期表（周期为

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

合并所有解

x = \frac{2 π}{3} + πn

作图

流行的例子

solvefor b,(sin(65)}{14}=\frac{sin(b))/9

so l v e f or b, \frac{sin ( 6 5 ^{\circ} )}{14} = \frac{sin ( b )}{9}

4sin(θ)=sin(2θ)

4 sin (θ) = sin (2 θ)

2sin(3x+(3pi)/2)=1

2 sin (3 x + \frac{3 π}{2}) = 1

n=(2tan^3(2θ)-1)^{1/2}

n = (2 tan^{3} (2 θ) - 1)^{\frac{1}{2}}

0 = 2 sin (x + 1) + 1