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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 4 5^{\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^{3} (x) - tan^{2} (x) - 3 tan (x) + 3 = 0

用替代法求解

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

令

: tan (x) = u

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

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

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

因式分解

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

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

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

从

- 3 u + 3

分解出因式

- 3

:

- 3 (u - 1)

- 3 u + 3

因式分解出通项

- 3

= - 3 (u - 1)

从

u^{3} - u^{2}

分解出因式

u^{2}

:

u^{2} (u - 1)

u^{3} - u^{2}

使用指数法则:

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

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

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

因式分解出通项

u^{2}

= u^{2} (u - 1)

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

因式分解出通项

u - 1

= (u - 1) (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 - 1) (u + 3) (u - 3)

(u - 1) (u + 3) (u - 3) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u - 1 = 0 or u + 3 = 0 or u - 3 = 0

解

u - 1 = 0 : u = 1

u - 1 = 0

将

1

到右边

u - 1 = 0

两边加上

1

u - 1 + 1 = 0 + 1

化简

u = 1

u = 1

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 = 1, u = - 3, u = 3

u = tan (x)

代回

tan (x) = 1, tan (x) = - 3, tan (x) = 3

tan (x) = 1, tan (x) = - 3, tan (x) = 3

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

tan (x) = 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}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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

作图

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