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受欢迎的三角函数 >

tan^2(x/2)-tan(x/2)-2=0

解答

tan^{2} (\frac{x}{2}) - tan (\frac{x}{2}) - 2 = 0

解答

x = 2 \cdot 1.10714 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

+1

度数

x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

tan^{2} (\frac{x}{2}) - tan (\frac{x}{2}) - 2 = 0

用替代法求解

tan^{2} (\frac{x}{2}) - tan (\frac{x}{2}) - 2 = 0

令

: tan (\frac{x}{2}) = u

u^{2} - u - 2 = 0

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

u^{2} - u - 2 = 0

使用求根公式求解

u^{2} - u - 2 = 0

二次方程求根公式:

若

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

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

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

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

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

使用法则

- (- a) = a

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

(- 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 1 \cdot 2 = 8

4 \cdot 1 \cdot 2

数字相乘:

4 \cdot 1 \cdot 2 = 8

= 8

= 1 + 8

数字相加:

1 + 8 = 9

= 9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相加:

1 + 3 = 4

= \frac{4}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

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

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

使用法则

- (- a) = a

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

数字相减:

1 - 3 = - 2

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 2}{2}

使用分式法则:

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

= - \frac{2}{2}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = 2, u = - 1

u = tan (\frac{x}{2})

代回

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

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

tan (\frac{x}{2}) = 2 : x = 2 arctan (2) + 2 πn

tan (\frac{x}{2}) = 2

使用反三角函数性质

tan (\frac{x}{2}) = 2

tan (\frac{x}{2}) = 2

的通解

tan (x) = a \Rightarrow x = arctan (a) + πn

\frac{x}{2} = arctan (2) + πn

\frac{x}{2} = arctan (2) + πn

解

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

\frac{x}{2} = arctan (2) + πn

在两边乘以

2

\frac{x}{2} = arctan (2) + πn

在两边乘以

2

\frac{2 x}{2} = 2 arctan (2) + 2 πn

化简

x = 2 arctan (2) + 2 πn

x = 2 arctan (2) + 2 πn

x = 2 arctan (2) + 2 πn

tan (\frac{x}{2}) = - 1 : x = \frac{3 π}{2} + 2 πn

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

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

的通解

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

\frac{x}{2} = \frac{3 π}{4} + πn

\frac{x}{2} = \frac{3 π}{4} + πn

解

\frac{x}{2} = \frac{3 π}{4} + πn : x = \frac{3 π}{2} + 2 πn

\frac{x}{2} = \frac{3 π}{4} + πn

在两边乘以

2

\frac{x}{2} = \frac{3 π}{4} + πn

在两边乘以

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 πn

化简

\frac{2 x}{2} = 2 \cdot \frac{3 π}{4} + 2 πn

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

2 \cdot \frac{3 π}{4} + 2 πn : \frac{3 π}{2} + 2 πn

2 \cdot \frac{3 π}{4} + 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

分式相乘:

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

= \frac{3 π 2}{4}

数字相乘:

3 \cdot 2 = 6

= \frac{6 π}{4}

约分:

2

= \frac{3 π}{2}

= \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

合并所有解

x = 2 arctan (2) + 2 πn, x = \frac{3 π}{2} + 2 πn

以小数形式表示解

x = 2 \cdot 1.10714 \dots + 2 πn, x = \frac{3 π}{2} + 2 πn

作图

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