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受欢迎的三角函数 >

tan^2(θ)-3cot(θ)=0

解答

tan^{2} (θ) - 3 cot (θ) = 0

解答

θ = 0.96453 \dots + πn

+1

度数

θ = 55.26405 \dots^{\circ} + 18 0^{\circ} n

求解步骤

tan^{2} (θ) - 3 cot (θ) = 0

使用三角恒等式改写

tan^{2} (θ) - 3 cot (θ)

使用基本三角恒等式:

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

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

(\frac{1}{cot ( θ )})^{2} = \frac{1}{cot ^{2} ( θ )}

(\frac{1}{cot ( θ )})^{2}

使用指数法则:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cot ^{2} ( θ )}

使用法则

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( θ )}

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

\frac{1}{cot ^{2} ( θ )} - 3 cot (θ) = 0

用替代法求解

\frac{1}{cot ^{2} ( θ )} - 3 cot (θ) = 0

令

: cot (θ) = u

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

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

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

在两边乘以

u^{2}

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

在两边乘以

u^{2}

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

化简

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

- 3 u u^{2} : - 3 u^{3}

- 3 u u^{2}

使用指数法则:

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

u u^{2} = u^{1 + 2}

= - 3 u^{1 + 2}

数字相加:

1 + 2 = 3

= - 3 u^{3}

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

1 - 3 u^{3} = 0

1 - 3 u^{3} = 0

1 - 3 u^{3} = 0

解

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

1 - 3 u^{3} = 0

将

1

到右边

1 - 3 u^{3} = 0

两边减去

1

1 - 3 u^{3} - 1 = 0 - 1

化简

- 3 u^{3} = - 1

- 3 u^{3} = - 1

两边除以

- 3

- 3 u^{3} = - 1

两边除以

- 3

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

化简

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

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

对于

x^{3} = f (a)

解为

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

u = 3 \frac{1}{3}, u = 3 \frac{1}{3} \frac{- 1 + 3 i}{2}, u = 3 \frac{1}{3} \frac{- 1 - 3 i}{2}

化简

3 \frac{1}{3} \frac{- 1 + 3 i}{2} : - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2}

3 \frac{1}{3} \frac{- 1 + 3 i}{2}

分式相乘:

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

= \frac{( - 1 + 3 i ) 3 \frac{1}{3}}{2}

3 \frac{1}{3} = \frac{1}{3 3}

3 \frac{1}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 3}

使用法则

n 1 = 1

31 = 1

= \frac{1}{3 3}

= \frac{\frac{1}{3 3} ( - 1 + 3 i )}{2}

乘

(- 1 + 3 i) \frac{1}{3 3} : \frac{- 1 + 3 i}{3 3}

(- 1 + 3 i) \frac{1}{3 3}

分式相乘:

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

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

1 \cdot (- 1 + 3 i) = - 1 + 3 i

1 \cdot (- 1 + 3 i)

乘以:

1 \cdot (- 1 + 3 i) = (- 1 + 3 i)

= (- 1 + 3 i)

去除括号:

(- a) = - a

= - 1 + 3 i

= \frac{- 1 + 3 i}{3 3}

= \frac{\frac{- 1 + 3 i}{3 3}}{2}

使用分式法则:

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

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

\frac{- 1 + 3 i}{2 3 3}

有理化:

\frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

\frac{- 1 + 3 i}{2 3 3}

乘以共轭根式

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

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

33 \cdot 2 \cdot 3^{\frac{2}{3}} = 6

33 \cdot 2 \cdot 3^{\frac{2}{3}}

使用指数法则:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

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

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

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

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

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 3^{1}

使用法则

a^{1} = a

= 3

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

= \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

= \frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

将

\frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

改写成标准复数形式:

- \frac{3 ^{\frac{2}{3}}}{6} + \frac{6 3}{2} i

\frac{3 ^{\frac{2}{3}} ( - 1 + 3 i )}{6}

分解

6 : 2 \cdot 3

因式分解

6 = 2 \cdot 3

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

消掉

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

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

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 + 3 i}{2 \cdot 3 ^{- \frac{2}{3} + 1}}

数字相减:

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

= \frac{- 1 + 3 i}{2 \cdot 3 ^{\frac{1}{3}}}

= \frac{- 1 + 3 i}{2 \cdot 3 ^{\frac{1}{3}}}

3^{\frac{1}{3}} = 33

使用根式运算法则:

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

3^{\frac{1}{3}} = 33

= \frac{- 1 + 3 i}{2 3 3}

使用分式法则:

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

\frac{- 1 + 3 i}{2 3 3} = - \frac{1}{2 3 3} + \frac{3 i}{2 3 3}

= - \frac{1}{2 3 3} + \frac{3 i}{2 3 3}

消掉

\frac{3 i}{2 3 3} : \frac{6 3 i}{2}

\frac{3 i}{2 3 3}

消掉

\frac{3 i}{2 3 3} : \frac{6 3 i}{2}

\frac{3 i}{2 3 3}

使用根式运算法则:

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

33 = 3^{\frac{1}{3}}, 3 = 3^{\frac{1}{2}}

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

使用指数法则:

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

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

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

数字相减:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2}

使用根式运算法则:

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

3^{\frac{1}{6}} = 63

= \frac{6 3 i}{2}

= \frac{6 3 i}{2}

= - \frac{1}{2 3 3} + \frac{6 3 i}{2}

- \frac{1}{2 3 3} = - \frac{3 ^{\frac{2}{3}}}{6}

- \frac{1}{2 3 3}

乘以共轭根式

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

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

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

233 \cdot 3^{\frac{2}{3}} = 6

233 \cdot 3^{\frac{2}{3}}

使用指数法则:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

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

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

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

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

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 3^{1}

使用法则

a^{1} = a

= 3

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

= - \frac{3 ^{\frac{2}{3}}}{6}

= - \frac{3 ^{\frac{2}{3}}}{6} + \frac{6 3}{2} i

= - \frac{3 ^{\frac{2}{3}}}{6} + \frac{6 3}{2} i

化简

3 \frac{1}{3} \frac{- 1 - 3 i}{2} : - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2}

3 \frac{1}{3} \frac{- 1 - 3 i}{2}

分式相乘:

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

= \frac{( - 1 - 3 i ) 3 \frac{1}{3}}{2}

3 \frac{1}{3} = \frac{1}{3 3}

3 \frac{1}{3}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{3 1}{3 3}

使用法则

n 1 = 1

31 = 1

= \frac{1}{3 3}

= \frac{\frac{1}{3 3} ( - 1 - 3 i )}{2}

乘

(- 1 - 3 i) \frac{1}{3 3} : \frac{- 1 - 3 i}{3 3}

(- 1 - 3 i) \frac{1}{3 3}

分式相乘:

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

= \frac{1 \cdot ( - 1 - 3 i )}{3 3}

1 \cdot (- 1 - 3 i) = - 1 - 3 i

1 \cdot (- 1 - 3 i)

乘以:

1 \cdot (- 1 - 3 i) = (- 1 - 3 i)

= (- 1 - 3 i)

去除括号:

(- a) = - a

= - 1 - 3 i

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

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

使用分式法则:

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

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

\frac{- 1 - 3 i}{2 3 3}

有理化:

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

\frac{- 1 - 3 i}{2 3 3}

乘以共轭根式

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

= \frac{( - 1 - 3 i ) \cdot 3 ^{\frac{2}{3}}}{3 3 \cdot 2 \cdot 3 ^{\frac{2}{3}}}

33 \cdot 2 \cdot 3^{\frac{2}{3}} = 6

33 \cdot 2 \cdot 3^{\frac{2}{3}}

使用指数法则:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

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

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

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

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

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 3^{1}

使用法则

a^{1} = a

= 3

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

将

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

改写成标准复数形式:

- \frac{3 ^{\frac{2}{3}}}{6} - \frac{6 3}{2} i

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{6}

分解

6 : 2 \cdot 3

因式分解

6 = 2 \cdot 3

= \frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 \cdot 3}

消掉

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 \cdot 3} : \frac{- 1 - 3 i}{2 \cdot 3 ^{\frac{1}{3}}}

\frac{3 ^{\frac{2}{3}} ( - 1 - 3 i )}{2 \cdot 3}

使用指数法则:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{- 1 - 3 i}{2 \cdot 3 ^{- \frac{2}{3} + 1}}

数字相减:

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

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

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

3^{\frac{1}{3}} = 33

使用根式运算法则:

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

3^{\frac{1}{3}} = 33

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

使用分式法则:

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

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

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

消掉

\frac{3 i}{2 3 3} : \frac{6 3 i}{2}

\frac{3 i}{2 3 3}

消掉

\frac{3 i}{2 3 3} : \frac{6 3 i}{2}

\frac{3 i}{2 3 3}

使用根式运算法则:

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

33 = 3^{\frac{1}{3}}, 3 = 3^{\frac{1}{2}}

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

使用指数法则:

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

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

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

数字相减:

\frac{1}{2} - \frac{1}{3} = \frac{1}{6}

= \frac{3 ^{\frac{1}{6}} i}{2}

使用根式运算法则:

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

3^{\frac{1}{6}} = 63

= \frac{6 3 i}{2}

= \frac{6 3 i}{2}

= - \frac{1}{2 3 3} - \frac{6 3 i}{2}

- \frac{1}{2 3 3} = - \frac{3 ^{\frac{2}{3}}}{6}

- \frac{1}{2 3 3}

乘以共轭根式

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

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

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

233 \cdot 3^{\frac{2}{3}} = 6

233 \cdot 3^{\frac{2}{3}}

使用指数法则:

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

3^{\frac{2}{3}} 33 = 3^{\frac{2}{3}} \cdot 3^{\frac{1}{3}} = 3^{\frac{2}{3} + \frac{1}{3}}

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

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

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

合并分式

\frac{2}{3} + \frac{1}{3} : 1

使用法则

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

= \frac{2 + 1}{3}

数字相加:

2 + 1 = 3

= \frac{3}{3}

使用法则

\frac{a}{a} = 1

= 1

= 3^{1}

使用法则

a^{1} = a

= 3

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

= - \frac{3 ^{\frac{2}{3}}}{6}

= - \frac{3 ^{\frac{2}{3}}}{6} - \frac{6 3}{2} i

= - \frac{3 ^{\frac{2}{3}}}{6} - \frac{6 3}{2} i

u = 3 \frac{1}{3}, u = - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2}, u = - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2}

u = 3 \frac{1}{3}, u = - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2}, u = - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2}

验证解

找到无定义的点（奇点）:

u = 0

取

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

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = 3 \frac{1}{3}, u = - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2}, u = - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2}

u = cot (θ)

代回

cot (θ) = 3 \frac{1}{3}, cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2}, cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2}

cot (θ) = 3 \frac{1}{3}, cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2}, cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2}

cot (θ) = 3 \frac{1}{3} : θ = arccot (3 \frac{1}{3}) + πn

cot (θ) = 3 \frac{1}{3}

使用反三角函数性质

cot (θ) = 3 \frac{1}{3}

cot (θ) = 3 \frac{1}{3}

的通解

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

θ = arccot (3 \frac{1}{3}) + πn

θ = arccot (3 \frac{1}{3}) + πn

cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2} :

无解

cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} + i \frac{6 3}{2}

无解

cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2} :

无解

cot (θ) = - \frac{3 ^{\frac{2}{3}}}{6} - i \frac{6 3}{2}

无解

合并所有解

θ = arccot (3 \frac{1}{3}) + πn

以小数形式表示解

θ = 0.96453 \dots + πn

作图

流行的例子

solvefor a,cos(ax)+(b/x)sin(ax)=0

so l v e f or a, cos (a x) + (\frac{b}{x}) sin (a x) = 0

cos^4(a)+cos^2(a)+sin^2(a)+sin^2(a)=1

cos^{4} (a) + cos^{2} (a) + sin^{2} (a) + sin^{2} (a) = 1

cos^3(x)-2sin(x)-0.7=0

cos^{3} (x) - 2 sin (x) - 0.7 = 0

solvefor y=cos(x),x

so l v e f or y = cos (x), x

tan(θ)= 5/(5sqrt(3))

tan (θ) = \frac{5}{5 3}