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受欢迎的三角函数 >

7tan(3x)-21tan(x)=0

解答

7 tan (3 x) - 21 tan (x) = 0

解答

x = πn

+1

度数

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

求解步骤

7 tan (3 x) - 21 tan (x) = 0

使用三角恒等式改写

- 21 tan (x) + 7 tan (3 x)

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

tan (3 x)

使用三角恒等式改写

tan (3 x)

改写为

= tan (2 x + x)

使用角和恒等式:

tan (s + t) = \frac{tan ( s ) + tan ( t )}{1 - tan ( s ) tan ( t )}

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

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

使用倍角公式:

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

= \frac{\frac{2 t a n ( x )}{1 - t a n ^{2} ( x )} + tan ( x )}{1 - \frac{2 t a n ( x )}{1 - t a n ^{2} ( x )} tan ( x )}

化简

\frac{\frac{2 t a n ( x )}{1 - t a n ^{2} ( x )} + tan ( x )}{1 - \frac{2 t a n ( x )}{1 - t a n ^{2} ( x )} tan ( x )} : \frac{3 tan ( x ) - tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )}

\frac{\frac{2 t a n ( x )}{1 - t a n ^{2} ( x )} + tan ( x )}{1 - \frac{2 t a n ( x )}{1 - t a n ^{2} ( x )} tan ( x )}

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

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

分式相乘:

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

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

2 tan (x) tan (x) = 2 tan^{2} (x)

2 tan (x) tan (x)

使用指数法则:

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

tan (x) tan (x) = tan^{1 + 1} (x)

= 2 tan^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 2 tan^{2} (x)

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

= \frac{\frac{2 t a n ( x )}{- t a n ^{2} ( x ) + 1} + tan ( x )}{1 - \frac{2 t a n ^{2} ( x )}{- t a n ^{2} ( x ) + 1}}

化简

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

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

将项转换为分式:

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

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

因为分母相等，所以合并分式:

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

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

乘开

2 tan (x) + tan (x) (1 - tan^{2} (x)) : 3 tan (x) - tan^{3} (x)

2 tan (x) + tan (x) (1 - tan^{2} (x))

乘开

tan (x) (1 - tan^{2} (x)) : tan (x) - tan^{3} (x)

tan (x) (1 - tan^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = tan (x), b = 1, c = tan^{2} (x)

= tan (x) 1 - tan (x) tan^{2} (x)

= 1 tan (x) - tan^{2} (x) tan (x)

化简

1 \cdot tan (x) - tan^{2} (x) tan (x) : tan (x) - tan^{3} (x)

1 tan (x) - tan^{2} (x) tan (x)

1 \cdot tan (x) = tan (x)

1 tan (x)

乘以:

1 \cdot tan (x) = tan (x)

= tan (x)

tan^{2} (x) tan (x) = tan^{3} (x)

tan^{2} (x) tan (x)

使用指数法则:

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

tan^{2} (x) tan (x) = tan^{2 + 1} (x)

= tan^{2 + 1} (x)

数字相加:

2 + 1 = 3

= tan^{3} (x)

= tan (x) - tan^{3} (x)

= tan (x) - tan^{3} (x)

= 2 tan (x) + tan (x) - tan^{3} (x)

同类项相加:

2 tan (x) + tan (x) = 3 tan (x)

= 3 tan (x) - tan^{3} (x)

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

= \frac{\frac{3 t a n ( x ) - t a n ^{3} ( x )}{1 - t a n ^{2} ( x )}}{1 - \frac{2 t a n ^{2} ( x )}{- t a n ^{2} ( x ) + 1}}

使用分式法则:

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

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

化简

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

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

将项转换为分式:

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

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

因为分母相等，所以合并分式:

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

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

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

1 (1 - tan^{2} (x)) - 2 tan^{2} (x)

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

1 (1 - tan^{2} (x))

乘以:

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

= 1 - tan^{2} (x)

去除括号:

(a) = a

= 1 - tan^{2} (x)

= 1 - tan^{2} (x) - 2 tan^{2} (x)

同类项相加:

- tan^{2} (x) - 2 tan^{2} (x) = - 3 tan^{2} (x)

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

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

= \frac{3 tan ( x ) - tan ^{3} ( x )}{\frac{- 3 t a n ^{2} ( x ) + 1}{- t a n ^{2} ( x ) + 1} ( - tan ^{2} ( x ) + 1 )}

乘

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

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

分式相乘:

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

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

约分:

1 - tan^{2} (x)

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

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

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

= - 21 tan (x) + 7 \cdot \frac{3 tan ( x ) - tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )}

化简

- 21 tan (x) + 7 \cdot \frac{3 tan ( x ) - tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )} : \frac{56 tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )}

- 21 tan (x) + 7 \cdot \frac{3 tan ( x ) - tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )}

乘

7 \cdot \frac{3 tan ( x ) - tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )} : \frac{7 ( 3 tan ( x ) - tan ^{3} ( x ) )}{1 - 3 tan ^{2} ( x )}

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

分式相乘:

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

= \frac{( 3 tan ( x ) - tan ^{3} ( x ) ) \cdot 7}{1 - 3 tan ^{2} ( x )}

= - 21 tan (x) + \frac{7 ( 3 tan ( x ) - tan ^{3} ( x ) )}{- 3 tan ^{2} ( x ) + 1}

将项转换为分式:

21 tan (x) = \frac{21 tan ( x ) ( 1 - 3 tan ^{2} ( x ) )}{1 - 3 tan ^{2} ( x )}

= \frac{( 3 tan ( x ) - tan ^{3} ( x ) ) \cdot 7}{1 - 3 tan ^{2} ( x )} - \frac{21 tan ( x ) ( 1 - 3 tan ^{2} ( x ) )}{1 - 3 tan ^{2} ( x )}

因为分母相等，所以合并分式:

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

= \frac{( 3 tan ( x ) - tan ^{3} ( x ) ) \cdot 7 - 21 tan ( x ) ( 1 - 3 tan ^{2} ( x ) )}{1 - 3 tan ^{2} ( x )}

乘开

(3 tan (x) - tan^{3} (x)) \cdot 7 - 21 tan (x) (1 - 3 tan^{2} (x)) : 56 tan^{3} (x)

(3 tan (x) - tan^{3} (x)) \cdot 7 - 21 tan (x) (1 - 3 tan^{2} (x))

= 7 (3 tan (x) - tan^{3} (x)) - 21 tan (x) (1 - 3 tan^{2} (x))

乘开

7 (3 tan (x) - tan^{3} (x)) : 21 tan (x) - 7 tan^{3} (x)

7 (3 tan (x) - tan^{3} (x))

使用分配律:

a (b - c) = ab - a c

a = 7, b = 3 tan (x), c = tan^{3} (x)

= 7 \cdot 3 tan (x) - 7 tan^{3} (x)

数字相乘:

7 \cdot 3 = 21

= 21 tan (x) - 7 tan^{3} (x)

= 21 tan (x) - 7 tan^{3} (x) - 21 tan (x) (1 - 3 tan^{2} (x))

乘开

- 21 tan (x) (1 - 3 tan^{2} (x)) : - 21 tan (x) + 63 tan^{3} (x)

- 21 tan (x) (1 - 3 tan^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 21 tan (x), b = 1, c = 3 tan^{2} (x)

= - 21 tan (x) \cdot 1 - (- 21 tan (x)) \cdot 3 tan^{2} (x)

使用加减运算法则

- (- a) = a

= - 21 \cdot 1 \cdot tan (x) + 21 \cdot 3 tan^{2} (x) tan (x)

化简

- 21 \cdot 1 \cdot tan (x) + 21 \cdot 3 tan^{2} (x) tan (x) : - 21 tan (x) + 63 tan^{3} (x)

- 21 \cdot 1 \cdot tan (x) + 21 \cdot 3 tan^{2} (x) tan (x)

21 \cdot 1 \cdot tan (x) = 21 tan (x)

21 \cdot 1 \cdot tan (x)

数字相乘:

21 \cdot 1 = 21

= 21 tan (x)

21 \cdot 3 tan^{2} (x) tan (x) = 63 tan^{3} (x)

21 \cdot 3 tan^{2} (x) tan (x)

数字相乘:

21 \cdot 3 = 63

= 63 tan^{2} (x) tan (x)

使用指数法则:

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

tan^{2} (x) tan (x) = tan^{2 + 1} (x)

= 63 tan^{2 + 1} (x)

数字相加:

2 + 1 = 3

= 63 tan^{3} (x)

= - 21 tan (x) + 63 tan^{3} (x)

= - 21 tan (x) + 63 tan^{3} (x)

= 21 tan (x) - 7 tan^{3} (x) - 21 tan (x) + 63 tan^{3} (x)

化简

21 tan (x) - 7 tan^{3} (x) - 21 tan (x) + 63 tan^{3} (x) : 56 tan^{3} (x)

21 tan (x) - 7 tan^{3} (x) - 21 tan (x) + 63 tan^{3} (x)

同类项相加:

- 7 tan^{3} (x) + 63 tan^{3} (x) = 56 tan^{3} (x)

= 21 tan (x) + 56 tan^{3} (x) - 21 tan (x)

同类项相加:

21 tan (x) - 21 tan (x) = 0

= 56 tan^{3} (x)

= 56 tan^{3} (x)

= \frac{56 tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )}

= \frac{56 tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )}

\frac{56 tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )} = 0

用替代法求解

\frac{56 tan ^{3} ( x )}{1 - 3 tan ^{2} ( x )} = 0

令

: tan (x) = u

\frac{56 u ^{3}}{1 - 3 u ^{2}} = 0

\frac{56 u ^{3}}{1 - 3 u ^{2}} = 0 : u = 0

\frac{56 u ^{3}}{1 - 3 u ^{2}} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

56 u^{3} = 0

解

56 u^{3} = 0 : u = 0

56 u^{3} = 0

两边除以

56

56 u^{3} = 0

两边除以

56

56 u^{3} = 0

两边除以

56

\frac{56 u ^{3}}{56} = \frac{0}{56}

化简

u^{3} = 0

u^{3} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

u = 0

验证解

找到无定义的点（奇点）:

u = \frac{1}{3}, u = - \frac{1}{3}

取

\frac{56 u ^{3}}{1 - 3 u ^{2}}

的分母，令其等于零

解

1 - 3 u^{2} = 0 : u = \frac{1}{3}, u = - \frac{1}{3}

1 - 3 u^{2} = 0

将

1

到右边

1 - 3 u^{2} = 0

两边减去

1

1 - 3 u^{2} - 1 = 0 - 1

化简

- 3 u^{2} = - 1

- 3 u^{2} = - 1

两边除以

- 3

- 3 u^{2} = - 1

两边除以

- 3

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

化简

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

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

对于

x^{2} = f (a)

解为

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

u = \frac{1}{3}, u = - \frac{1}{3}

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

\frac{1}{3}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= \frac{1}{3}

使用根式运算法则:

1 = 1

1 = 1

= \frac{1}{3}

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

- \frac{1}{3}

使用根式运算法则:

\frac{a}{b} = \frac{a}{b}, a \geq 0, b \geq 0

= - \frac{1}{3}

使用根式运算法则:

1 = 1

1 = 1

= - \frac{1}{3}

u = \frac{1}{3}, u = - \frac{1}{3}

以下点无定义

u = \frac{1}{3}, u = - \frac{1}{3}

将不在定义域的点与解相综合:

u = 0

u = tan (x)

代回

tan (x) = 0

tan (x) = 0

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

合并所有解

x = πn

作图

流行的例子

cos (A) = - \frac{3}{25}

6sin(2x)=3cos(30)

6 sin (2 x) = 3 cos (3 0^{\circ})

cos (y) = - 1

cot (x) = - \frac{2}{3}

tan((3θ)/4)=(-sqrt(3))/3

tan (\frac{3 θ}{4}) = \frac{- 3}{3}