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受欢迎的三角函数 >

2tan^3(x)-2/3 tan(x)=0

解答

2 tan^{3} (x) - \frac{2}{3} tan (x) = 0

解答

x = πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

求解步骤

2 tan^{3} (x) - \frac{2}{3} tan (x) = 0

用替代法求解

2 tan^{3} (x) - \frac{2}{3} tan (x) = 0

令

: tan (x) = u

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

2 u^{3} - \frac{2}{3} u = 0 : u = 0, u = - \frac{3}{3}, u = \frac{3}{3}

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

因式分解

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

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

因式分解出通项

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

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

使用指数法则:

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

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

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

因式分解出通项

u

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

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

分解

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

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

将

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

改写为

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

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

使用根式运算法则:

a = (a)^{2}

2 = (2)^{2}

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

使用根式运算法则:

a = (a)^{2}

\frac{2}{3} = (\frac{2}{3})^{2}

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

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(2)^{2} u^{2} = (2 u)^{2}

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

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

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

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

u (2 u + \frac{2}{3}) (2 u - \frac{2}{3}) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 2 u + \frac{2}{3} = 0 or 2 u - \frac{2}{3} = 0

解

2 u + \frac{2}{3} = 0 : u = - \frac{3}{3}

2 u + \frac{2}{3} = 0

将

\frac{2}{3}

到右边

2 u + \frac{2}{3} = 0

两边减去

\frac{2}{3}

2 u + \frac{2}{3} - \frac{2}{3} = 0 - \frac{2}{3}

化简

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

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

两边除以

2

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

两边除以

2

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

化简

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

化简

\frac{2 u}{2} : u

\frac{2 u}{2}

约分:

2

= u

化简

\frac{- \frac{2}{3}}{2} : - \frac{3}{3}

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

使用分式法则:

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

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

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

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

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

使用分式法则:

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

= - \frac{2}{3 2}

约分:

2

= - \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}

u = - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

解

2 u - \frac{2}{3} = 0 : u = \frac{3}{3}

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

将

\frac{2}{3}

到右边

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

两边加上

\frac{2}{3}

2 u - \frac{2}{3} + \frac{2}{3} = 0 + \frac{2}{3}

化简

2 u = \frac{2}{3}

2 u = \frac{2}{3}

两边除以

2

2 u = \frac{2}{3}

两边除以

2

\frac{2 u}{2} = \frac{\frac{2}{3}}{2}

化简

\frac{2 u}{2} = \frac{\frac{2}{3}}{2}

化简

\frac{2 u}{2} : u

\frac{2 u}{2}

约分:

2

= u

化简

\frac{\frac{2}{3}}{2} : \frac{3}{3}

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

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

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

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

使用分式法则:

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

= \frac{2}{3 2}

约分:

2

= \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}

u = \frac{3}{3}

u = \frac{3}{3}

u = \frac{3}{3}

解为

u = 0, u = - \frac{3}{3}, u = \frac{3}{3}

u = tan (x)

代回

tan (x) = 0, tan (x) = - \frac{3}{3}, tan (x) = \frac{3}{3}

tan (x) = 0, tan (x) = - \frac{3}{3}, tan (x) = \frac{3}{3}

tan (x) = 0 : x = πn

tan (x) = 0

tan (x) = 0

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 = 0 + πn

x = 0 + πn

解

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

tan (x) = - \frac{3}{3} : x = \frac{5 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{5 π}{6} + πn

x = \frac{5 π}{6} + 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 = πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

作图

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