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受欢迎的三角函数 >

2tan^3(x)-6tan(x)=0,0<= x<= pi

解答

2 tan^{3} (x) - 6 tan (x) = 0, 0 \leq x \leq π

解答

x = 0, x = π, x = \frac{2 π}{3}, x = \frac{π}{3}

+1

度数

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

求解步骤

2 tan^{3} (x) - 6 tan (x) = 0, 0 \leq x \leq π

用替代法求解

2 tan^{3} (x) - 6 tan (x) = 0

令

: tan (x) = u

2 u^{3} - 6 u = 0

2 u^{3} - 6 u = 0 : u = 0, u = - 3, u = 3

2 u^{3} - 6 u = 0

因式分解

2 u^{3} - 6 u : 2 u (u + 3) (u - 3)

2 u^{3} - 6 u

因式分解出通项

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

2 u^{3} - 6 u

使用指数法则:

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

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

= 2 u^{2} u - 6 u

将

6

改写为

2 \cdot 3

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

因式分解出通项

2 u

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

= 2 u (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)

= 2 u (u + 3) (u - 3)

2 u (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 + 3 = 0 or u - 3 = 0

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, u = 3

u = tan (x)

代回

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

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

tan (x) = 0, 0 \leq x \leq π : x = 0, x = π

tan (x) = 0, 0 \leq x \leq π

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

在

0 \leq x \leq π

范围内的解

x = 0, x = π

tan (x) = - 3, 0 \leq x \leq π : x = \frac{2 π}{3}

tan (x) = - 3, 0 \leq x \leq π

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

在

0 \leq x \leq π

范围内的解

x = \frac{2 π}{3}

tan (x) = 3, 0 \leq x \leq π : x = \frac{π}{3}

tan (x) = 3, 0 \leq x \leq π

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

在

0 \leq x \leq π

范围内的解

x = \frac{π}{3}

合并所有解

x = 0, x = π, x = \frac{2 π}{3}, x = \frac{π}{3}

作图

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