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受欢迎的三角函数 >

tan^5(x)-9tan(x)=0

解答

tan^{5} (x) - 9 tan (x) = 0

解答

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

+1

度数

x = 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

求解步骤

tan^{5} (x) - 9 tan (x) = 0

用替代法求解

tan^{5} (x) - 9 tan (x) = 0

令

: tan (x) = u

u^{5} - 9 u = 0

u^{5} - 9 u = 0 : u = 0, u = 3 i, u = - 3 i, u = - 3, u = 3

u^{5} - 9 u = 0

因式分解

u^{5} - 9 u : u (u^{2} + 3) (u + 3) (u - 3)

u^{5} - 9 u

因式分解出通项

u : u (u^{4} - 9)

u^{5} - 9 u

使用指数法则:

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

u^{5} = u^{4} u

= u^{4} u - 9 u

因式分解出通项

u

= u (u^{4} - 9)

= u (u^{4} - 9)

分解

u^{4} - 9 : (u^{2} + 3) (u + 3) (u - 3)

u^{4} - 9

将

u^{4} - 9

改写为

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

u^{4} - 9

将

9

改写为

3^{2}

= u^{4} - 3^{2}

使用指数法则:

a^{b c} = (a^{b})^{c}

u^{4} = (u^{2})^{2}

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

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

使用平方差公式:

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

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

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

分解

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

u^{2} - 3

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

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

使用平方差公式:

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

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

= (u + 3) (u - 3)

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

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

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

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

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

解

u^{2} + 3 = 0 : u = 3 i, u = - 3 i

u^{2} + 3 = 0

将

3

到右边

u^{2} + 3 = 0

两边减去

3

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

化简

u^{2} = - 3

u^{2} = - 3

对于

x^{2} = f (a)

解为

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

u = - 3, u = - - 3

化简

- 3 : 3 i

- 3

使用根式运算法则:

- a = - 1 a

- 3 = - 1 3

= - 1 3

使用虚数运算法则:

- 1 = i

= 3 i

化简

- - 3 : - 3 i

- - 3

化简

- 3 : 3 i

- 3

使用根式运算法则:

- a = - 1 a

- 3 = - 1 3

= - 1 3

使用虚数运算法则:

- 1 = i

= 3 i

= - 3 i

u = 3 i, u = - 3 i

解

u + 3 = 0 : u = - 3

u + 3 = 0

将

3

到右边

u + 3 = 0

两边减去

3

u + 3 - 3 = 0 - 3

化简

u = - 3

u = - 3

解

u - 3 = 0 : u = 3

u - 3 = 0

将

3

到右边

u - 3 = 0

两边加上

3

u - 3 + 3 = 0 + 3

化简

u = 3

u = 3

解为

u = 0, u = 3 i, u = - 3 i, u = - 3, u = 3

u = tan (x)

代回

tan (x) = 0, tan (x) = 3 i, tan (x) = - 3 i, tan (x) = - 3, tan (x) = 3

tan (x) = 0, tan (x) = 3 i, tan (x) = - 3 i, tan (x) = - 3, tan (x) = 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) = 3 i :

无解

tan (x) = 3 i

无解

tan (x) = - 3 i :

无解

tan (x) = - 3 i

无解

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) = 3 : x = \frac{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{π}{3} + πn

x = \frac{π}{3} + πn

合并所有解

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

作图

流行的例子

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(tan(θ)-1)(cos(θ)+1)=0

(tan (θ) - 1) (cos (θ) + 1) = 0

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sin^{2} (θ) = cos^{2} (θ)

solvefor x,sin(x)=0

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

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