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受欢迎的三角函数 >

tan(x)=2-cot(x)

解答

tan (x) = 2 - cot (x)

解答

x = \frac{π}{4} + πn

+1

度数

x = 4 5^{\circ} + 18 0^{\circ} n

求解步骤

tan (x) = 2 - cot (x)

两边减去

2 - cot (x)

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

使用三角恒等式改写

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

使用基本三角恒等式:

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

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

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

用替代法求解

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

令

: cot (x) = u

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

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

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

在两边乘以

u

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

在两边乘以

u

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

化简

- 2 u + uu + \frac{1}{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{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

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

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

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

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

解

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = - 2, c = 1

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

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

(- 2)^{2} - 4 \cdot 1 \cdot 1 = 0

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

使用指数法则:

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

若

n

是偶数

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

= 2^{2} - 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 2^{2} - 4

2^{2} = 4

= 4 - 4

数字相减:

4 - 4 = 0

= 0

u_{1, 2} = \frac{- ( - 2 ) \pm 0}{2 \cdot 1}

u = \frac{- ( - 2 )}{2 \cdot 1}

\frac{- ( - 2 )}{2 \cdot 1} = 1

\frac{- ( - 2 )}{2 \cdot 1}

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

u = 1

二次方程组的解是:

u = 1

u = 1

验证解

找到无定义的点（奇点）:

u = 0

取

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

u = cot (x)

代回

cot (x) = 1

cot (x) = 1

cot (x) = 1 : x = \frac{π}{4} + πn

cot (x) = 1

cot (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

合并所有解

x = \frac{π}{4} + πn

作图

流行的例子

sin(θ)=cos(37)

cot(x)-cot(x)csc(x)=0

12cos^2(x)-7cos(x)+1=0

tan(x/2)tan(x)=-3