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受欢迎的三角函数 >

(2tan(x))/(1-tan^2(x))=sqrt(3)

解答

\frac{2 tan ( x )}{1 - tan ^{2} ( x )} = 3

解答

x = \frac{2 π}{3} + πn, x = \frac{π}{6} + πn

+1

度数

x = 12 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

求解步骤

\frac{2 tan ( x )}{1 - tan ^{2} ( x )} = 3

用替代法求解

\frac{2 tan ( x )}{1 - tan ^{2} ( x )} = 3

令

: tan (x) = u

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

\frac{2 u}{1 - u ^{2}} = 3 : u = - 3, u = \frac{3}{3}

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

在两边乘以

1 - u^{2}

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

在两边乘以

1 - u^{2}

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

化简

2 u = 3 (1 - u^{2})

2 u = 3 (1 - u^{2})

解

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

2 u = 3 (1 - u^{2})

展开

3 (1 - u^{2}) : 3 - 3 u^{2}

3 (1 - u^{2})

使用分配律:

a (b - c) = ab - a c

a = 3, b = 1, c = u^{2}

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

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

乘以:

1 \cdot 3 = 3

= 3 - 3 u^{2}

2 u = 3 - 3 u^{2}

交换两边

3 - 3 u^{2} = 2 u

将

2 u

para o lado esquerdo

3 - 3 u^{2} = 2 u

两边减去

2 u

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

化简

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = - 3, b = - 2, c = 3

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

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

(- 2)^{2} - 4 (- 3) 3 = 4

(- 2)^{2} - 4 (- 3) 3

使用法则

- (- a) = a

= (- 2)^{2} + 433

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

(- 2)^{2}

使用指数法则:

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

若

n

是偶数

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

= 2^{2}

433 = 12

433

使用根式运算法则:

a a = a

33 = 3

= 4 \cdot 3

数字相乘:

4 \cdot 3 = 12

= 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

数字相加:

4 + 12 = 16

= 16

因式分解数字:

16 = 4^{2}

= 4^{2}

使用根式运算法则:

n a^{n} = a

4^{2} = 4

= 4

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

将解分隔开

u_{1} = \frac{- ( - 2 ) + 4}{2 ( - 3 )}, u_{2} = \frac{- ( - 2 ) - 4}{2 ( - 3 )}

u = \frac{- ( - 2 ) + 4}{2 ( - 3 )} : - 3

\frac{- ( - 2 ) + 4}{2 ( - 3 )}

去除括号:

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

= \frac{2 + 4}{- 2 3}

数字相加:

2 + 4 = 6

= \frac{6}{- 2 3}

使用分式法则:

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

= - \frac{6}{2 3}

数字相除:

\frac{6}{2} = 3

= \frac{3}{3}

使用根式运算法则:

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

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

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

使用指数法则:

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

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

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

数字相减:

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

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

使用根式运算法则:

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

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

= - 3

u = \frac{- ( - 2 ) - 4}{2 ( - 3 )} : \frac{3}{3}

\frac{- ( - 2 ) - 4}{2 ( - 3 )}

去除括号:

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

= \frac{2 - 4}{- 2 3}

数字相减:

2 - 4 = - 2

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

使用分式法则:

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

= \frac{2}{2 3}

数字相除:

\frac{2}{2} = 1

= \frac{1}{3}

\frac{1}{3}

有理化:

\frac{3}{3}

\frac{1}{3}

乘以共轭根式

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

二次方程组的解是:

u = - 3, u = \frac{3}{3}

u = - 3, u = \frac{3}{3}

验证解

找到无定义的点（奇点）:

u = 1, u = - 1

取

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

的分母，令其等于零

解

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

1 - u^{2} = 0

将

1

到右边

1 - u^{2} = 0

两边减去

1

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

化简

- u^{2} = - 1

- u^{2} = - 1

两边除以

- 1

- u^{2} = - 1

两边除以

- 1

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

化简

u^{2} = 1

u^{2} = 1

对于

x^{2} = f (a)

解为

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

u = 1, u = - 1

1 = 1

1

使用根式运算法则:

1 = 1

= 1

- 1 = - 1

- 1

使用根式运算法则:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

以下点无定义

u = 1, u = - 1

将不在定义域的点与解相综合:

u = - 3, u = \frac{3}{3}

u = tan (x)

代回

tan (x) = - 3, tan (x) = \frac{3}{3}

tan (x) = - 3, tan (x) = \frac{3}{3}

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

tan (x) = - 3

tan (x) = - 3

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}

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

tan (x) = \frac{3}{3} : x = \frac{π}{6} + πn

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

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}

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

合并所有解

x = \frac{2 π}{3} + πn, x = \frac{π}{6} + πn

作图

流行的例子

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cos (θ) = \frac{5}{5 2}

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arccos (x) = arcsin (\frac{9}{41})

solvefor x,-ysin(x)=0

so l v e f or x, - y sin (x) = 0

2sin(x/2+pi/3)=sqrt(3)

2 sin (\frac{x}{2} + \frac{π}{3}) = 3