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3tan(x)-3cot(x)-1=0

解答

3 tan (x) - 3 cot (x) - 1 = 0

解答

x = 2.43876 \dots + πn, x = 0.86797 \dots + πn

+1

度数

x = 139.73116 \dots^{\circ} + 18 0^{\circ} n, x = 49.73116 \dots^{\circ} + 18 0^{\circ} n

求解步骤

3 tan (x) - 3 cot (x) - 1 = 0

使用三角恒等式改写

- 1 - 3 cot (x) + 3 tan (x)

使用基本三角恒等式:

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

= - 1 - 3 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 )}

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

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

用替代法求解

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

令

: cot (x) = u

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

- 1 + \frac{3}{u} - 3 u = 0 : u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

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

在两边乘以

u

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

在两边乘以

u

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

化简

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

化简

- 1 \cdot u : - u

- 1 \cdot u

乘以:

1 \cdot u = u

= - u

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

- 3 uu : - 3 u^{2}

- 3 uu

使用指数法则:

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

uu = u^{1 + 1}

= - 3 u^{1 + 1}

数字相加:

1 + 1 = 2

= - 3 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

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

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

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

解

- u + 3 - 3 u^{2} = 0 : u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 3, b = - 1, c = 3

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

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

(- 1)^{2} - 4 (- 3) \cdot 3 = 37

(- 1)^{2} - 4 (- 3) \cdot 3

使用法则

- (- a) = a

= (- 1)^{2} + 4 \cdot 3 \cdot 3

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

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

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 3 \cdot 3 = 36

4 \cdot 3 \cdot 3

数字相乘:

4 \cdot 3 \cdot 3 = 36

= 36

= 1 + 36

数字相加:

1 + 36 = 37

= 37

u_{1, 2} = \frac{- ( - 1 ) \pm 37}{2 ( - 3 )}

将解分隔开

u_{1} = \frac{- ( - 1 ) + 37}{2 ( - 3 )}, u_{2} = \frac{- ( - 1 ) - 37}{2 ( - 3 )}

u = \frac{- ( - 1 ) + 37}{2 ( - 3 )} : - \frac{1 + 37}{6}

\frac{- ( - 1 ) + 37}{2 ( - 3 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{1 + 37}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{1 + 37}{- 6}

使用分式法则:

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

= - \frac{1 + 37}{6}

u = \frac{- ( - 1 ) - 37}{2 ( - 3 )} : \frac{37 - 1}{6}

\frac{- ( - 1 ) - 37}{2 ( - 3 )}

去除括号:

(- a) = - a, - (- a) = a

= \frac{1 - 37}{- 2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{1 - 37}{- 6}

使用分式法则:

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

1 - 37 = - (37 - 1)

= \frac{37 - 1}{6}

二次方程组的解是:

u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

u = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

验证解

找到无定义的点（奇点）:

u = 0

取

- 1 + \frac{3}{u} - 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 = - \frac{1 + 37}{6}, u = \frac{37 - 1}{6}

u = cot (x)

代回

cot (x) = - \frac{1 + 37}{6}, cot (x) = \frac{37 - 1}{6}

cot (x) = - \frac{1 + 37}{6}, cot (x) = \frac{37 - 1}{6}

cot (x) = - \frac{1 + 37}{6} : x = arccot (- \frac{1 + 37}{6}) + πn

cot (x) = - \frac{1 + 37}{6}

使用反三角函数性质

cot (x) = - \frac{1 + 37}{6}

cot (x) = - \frac{1 + 37}{6}

的通解

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- \frac{1 + 37}{6}) + πn

x = arccot (- \frac{1 + 37}{6}) + πn

cot (x) = \frac{37 - 1}{6} : x = arccot (\frac{37 - 1}{6}) + πn

cot (x) = \frac{37 - 1}{6}

使用反三角函数性质

cot (x) = \frac{37 - 1}{6}

cot (x) = \frac{37 - 1}{6}

的通解

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (\frac{37 - 1}{6}) + πn

x = arccot (\frac{37 - 1}{6}) + πn

合并所有解

x = arccot (- \frac{1 + 37}{6}) + πn, x = arccot (\frac{37 - 1}{6}) + πn

以小数形式表示解

x = 2.43876 \dots + πn, x = 0.86797 \dots + πn

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