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受欢迎的三角函数 >

tan(2x)=2cot(2x)

解答

tan (2 x) = 2 cot (2 x)

解答

x = \frac{0.95531 \dots}{2} + \frac{πn}{2}, x = \frac{2.18627 \dots}{2} + \frac{πn}{2}

+1

度数

x = 27.36780 \dots^{\circ} + 9 0^{\circ} n, x = 62.63219 \dots^{\circ} + 9 0^{\circ} n

求解步骤

tan (2 x) = 2 cot (2 x)

两边减去

2 cot (2 x)

tan (2 x) - 2 cot (2 x) = 0

使用三角恒等式改写

tan (2 x) - 2 cot (2 x)

使用基本三角恒等式:

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

= \frac{1}{cot ( 2 x )} - 2 cot (2 x)

\frac{1}{cot ( 2 x )} - 2 cot (2 x) = 0

用替代法求解

\frac{1}{cot ( 2 x )} - 2 cot (2 x) = 0

令

: cot (2 x) = u

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

\frac{1}{u} - 2 u = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

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

在两边乘以

u

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

在两边乘以

u

\frac{1}{u} u - 2 uu = 0 \cdot u

化简

\frac{1}{u} u - 2 uu = 0 \cdot u

化简

\frac{1}{u} u : 1

\frac{1}{u} u

分式相乘:

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

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

约分:

u

= 1

化简

- 2 uu : - 2 u^{2}

- 2 uu

使用指数法则:

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

uu = u^{1 + 1}

= - 2 u^{1 + 1}

数字相加:

1 + 1 = 2

= - 2 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

1 - 2 u^{2} = 0

1 - 2 u^{2} = 0

1 - 2 u^{2} = 0

解

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

1 - 2 u^{2} = 0

将

1

到右边

1 - 2 u^{2} = 0

两边减去

1

1 - 2 u^{2} - 1 = 0 - 1

化简

- 2 u^{2} = - 1

- 2 u^{2} = - 1

两边除以

- 2

- 2 u^{2} = - 1

两边除以

- 2

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

化简

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

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

对于

x^{2} = f (a)

解为

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

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

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

验证解

找到无定义的点（奇点）:

u = 0

取

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

u = cot (2 x)

代回

cot (2 x) = \frac{1}{2}, cot (2 x) = - \frac{1}{2}

cot (2 x) = \frac{1}{2}, cot (2 x) = - \frac{1}{2}

cot (2 x) = \frac{1}{2} : x = \frac{arccot ( \frac{1}{2} )}{2} + \frac{πn}{2}

cot (2 x) = \frac{1}{2}

使用反三角函数性质

cot (2 x) = \frac{1}{2}

cot (2 x) = \frac{1}{2}

的通解

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

2 x = arccot (\frac{1}{2}) + πn

2 x = arccot (\frac{1}{2}) + πn

解

2 x = arccot (\frac{1}{2}) + πn : x = \frac{arccot ( \frac{1}{2} )}{2} + \frac{πn}{2}

2 x = arccot (\frac{1}{2}) + πn

两边除以

2

2 x = arccot (\frac{1}{2}) + πn

两边除以

2

\frac{2 x}{2} = \frac{arccot ( \frac{1}{2} )}{2} + \frac{πn}{2}

化简

x = \frac{arccot ( \frac{1}{2} )}{2} + \frac{πn}{2}

x = \frac{arccot ( \frac{1}{2} )}{2} + \frac{πn}{2}

x = \frac{arccot ( \frac{1}{2} )}{2} + \frac{πn}{2}

cot (2 x) = - \frac{1}{2} : x = \frac{arccot ( - \frac{1}{2} )}{2} + \frac{πn}{2}

cot (2 x) = - \frac{1}{2}

使用反三角函数性质

cot (2 x) = - \frac{1}{2}

cot (2 x) = - \frac{1}{2}

的通解

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

2 x = arccot (- \frac{1}{2}) + πn

2 x = arccot (- \frac{1}{2}) + πn

解

2 x = arccot (- \frac{1}{2}) + πn : x = \frac{arccot ( - \frac{1}{2} )}{2} + \frac{πn}{2}

2 x = arccot (- \frac{1}{2}) + πn

两边除以

2

2 x = arccot (- \frac{1}{2}) + πn

两边除以

2

\frac{2 x}{2} = \frac{arccot ( - \frac{1}{2} )}{2} + \frac{πn}{2}

化简

x = \frac{arccot ( - \frac{1}{2} )}{2} + \frac{πn}{2}

x = \frac{arccot ( - \frac{1}{2} )}{2} + \frac{πn}{2}

x = \frac{arccot ( - \frac{1}{2} )}{2} + \frac{πn}{2}

合并所有解

x = \frac{arccot ( \frac{1}{2} )}{2} + \frac{πn}{2}, x = \frac{arccot ( - \frac{1}{2} )}{2} + \frac{πn}{2}

以小数形式表示解

x = \frac{0.95531 \dots}{2} + \frac{πn}{2}, x = \frac{2.18627 \dots}{2} + \frac{πn}{2}

作图

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