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受欢迎的三角函数 >

sqrt(3)cot^2(x)-4cot(x)+sqrt(3)=0

解答

3 cot^{2} (x) - 4 cot (x) + 3 = 0

解答

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

+1

度数

x = 3 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

求解步骤

3 cot^{2} (x) - 4 cot (x) + 3 = 0

用替代法求解

3 cot^{2} (x) - 4 cot (x) + 3 = 0

令

: cot (x) = u

3 u^{2} - 4 u + 3 = 0

3 u^{2} - 4 u + 3 = 0 : u = 3, u = \frac{3}{3}

3 u^{2} - 4 u + 3 = 0

使用求根公式求解

3 u^{2} - 4 u + 3 = 0

二次方程求根公式:

若

a = 3, b = - 4, c = 3

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

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

(- 4)^{2} - 433 = 2

(- 4)^{2} - 433

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

(- 4)^{2}

使用指数法则:

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

若

n

是偶数

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

= 4^{2}

433 = 12

433

使用根式运算法则:

a a = a

33 = 3

= 4 \cdot 3

数字相乘:

4 \cdot 3 = 12

= 12

= 4^{2} - 12

4^{2} = 16

= 16 - 12

数字相减:

16 - 12 = 4

= 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

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

将解分隔开

u_{1} = \frac{- ( - 4 ) + 2}{2 3}, u_{2} = \frac{- ( - 4 ) - 2}{2 3}

u = \frac{- ( - 4 ) + 2}{2 3} : 3

\frac{- ( - 4 ) + 2}{2 3}

使用法则

- (- a) = a

= \frac{4 + 2}{2 3}

数字相加:

4 + 2 = 6

= \frac{6}{2 3}

数字相除:

\frac{6}{2} = 3

= \frac{3}{3}

使用根式运算法则:

n a = a^{\frac{1}{n}}

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

使用指数法则:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

数字相减:

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

= 3^{\frac{1}{2}}

使用根式运算法则:

a^{\frac{1}{n}} = n a

3^{\frac{1}{2}} = 3

= 3

u = \frac{- ( - 4 ) - 2}{2 3} : \frac{3}{3}

\frac{- ( - 4 ) - 2}{2 3}

使用法则

- (- a) = a

= \frac{4 - 2}{2 3}

数字相减:

4 - 2 = 2

= \frac{2}{2 3}

数字相除:

\frac{2}{2} = 1

= \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}

二次方程组的解是:

u = 3, u = \frac{3}{3}

u = cot (x)

代回

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

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

cot (x) = 3 : x = \frac{π}{6} + πn

cot (x) = 3

cot (x) = 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{π}{6} + πn

x = \frac{π}{6} + πn

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

x = \frac{π}{3} + πn

合并所有解

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

作图

流行的例子

3 cos^{2} (θ) - 1 = 0

sin(3x-pi/6)=-cos(3x-pi/6)

sin (3 x - \frac{π}{6}) = - cos (3 x - \frac{π}{6})

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