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受欢迎的三角函数 >

cot^3(x)+2cot^2(x)-3cot(x)-6=0

解答

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

解答

x = 2.67794 \dots + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

+1

度数

x = 153.43494 \dots^{\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

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

用替代法求解

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

令

: cot (x) = u

u^{3} + 2 u^{2} - 3 u - 6 = 0

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

u^{3} + 2 u^{2} - 3 u - 6 = 0

因式分解

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

u^{3} + 2 u^{2} - 3 u - 6

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

从

- 3 u - 6

分解出因式

- 3

:

- 3 (u + 2)

- 3 u - 6

将

6

改写为

3 \cdot 2

= - 3 u - 3 \cdot 2

因式分解出通项

- 3

= - 3 (u + 2)

从

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

分解出因式

u^{2}

:

u^{2} (u + 2)

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

使用指数法则:

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

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

= u u^{2} + 2 u^{2}

因式分解出通项

u^{2}

= u^{2} (u + 2)

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

因式分解出通项

u + 2

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

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

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

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

解

u + 2 = 0 : u = - 2

u + 2 = 0

将

2

到右边

u + 2 = 0

两边减去

2

u + 2 - 2 = 0 - 2

化简

u = - 2

u = - 2

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 = - 2, u = - 3, u = 3

u = cot (x)

代回

cot (x) = - 2, cot (x) = - 3, cot (x) = 3

cot (x) = - 2, cot (x) = - 3, cot (x) = 3

cot (x) = - 2 : x = arccot (- 2) + πn

cot (x) = - 2

使用反三角函数性质

cot (x) = - 2

cot (x) = - 2

的通解

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- 2) + πn

x = arccot (- 2) + πn

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

cot (x) = - 3

cot (x) = - 3

的通解

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

cot (x) = 3 : x = \frac{π}{6} + πn

cot (x) = 3

cot (x) = 3

的通解

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

合并所有解

x = arccot (- 2) + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

以小数形式表示解

x = 2.67794 \dots + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

作图

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