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受欢迎的三角函数 >

3tan(x)-cot(x)=0

解答

3 tan (x) - cot (x) = 0

解答

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

+1

度数

x = 3 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

求解步骤

3 tan (x) - cot (x) = 0

使用三角恒等式改写

- cot (x) + 3 tan (x)

使用基本三角恒等式:

tan (x) = \frac{1}{cot ( x )}

= - cot (x) + 3 \cdot \frac{1}{cot ( x )}

3 \cdot \frac{1}{cot ( x )} = \frac{3}{cot ( x )}

3 \cdot \frac{1}{cot ( x )}

分式相乘:

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

= \frac{1 \cdot 3}{cot ( x )}

数字相乘:

1 \cdot 3 = 3

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

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

- cot (x) + \frac{3}{cot ( x )} = 0

用替代法求解

- cot (x) + \frac{3}{cot ( x )} = 0

令

: cot (x) = u

- u + \frac{3}{u} = 0

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

- u + \frac{3}{u} = 0

在两边乘以

u

- u + \frac{3}{u} = 0

在两边乘以

u

- uu + \frac{3}{u} u = 0 \cdot u

化简

- uu + \frac{3}{u} u = 0 \cdot u

化简

- uu : - u^{2}

- uu

使用指数法则:

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

uu = u^{1 + 1}

= - u^{1 + 1}

数字相加:

1 + 1 = 2

= - u^{2}

化简

\frac{3}{u} u : 3

\frac{3}{u} u

分式相乘:

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

= \frac{3 u}{u}

约分:

u

= 3

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- u^{2} + 3 = 0

- u^{2} + 3 = 0

- u^{2} + 3 = 0

解

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

- u^{2} + 3 = 0

将

3

到右边

- u^{2} + 3 = 0

两边减去

3

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

化简

- u^{2} = - 3

- u^{2} = - 3

两边除以

- 1

- u^{2} = - 3

两边除以

- 1

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

化简

u^{2} = 3

u^{2} = 3

对于

x^{2} = f (a)

解为

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

u = 3, u = - 3

u = 3, u = - 3

验证解

找到无定义的点（奇点）:

u = 0

取

- u + \frac{3}{u}

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = 3, u = - 3

u = cot (x)

代回

cot (x) = 3, cot (x) = - 3

cot (x) = 3, cot (x) = - 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) = - 3 : x = \frac{5 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{5 π}{6} + πn

x = \frac{5 π}{6} + πn

合并所有解

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

作图

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