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受欢迎的三角函数 >

cot^4(x)-3cot^2(x)+1=0

解答

cot^{4} (x) - 3 cot^{2} (x) + 1 = 0

解答

x = 0.55357 \dots + πn, x = 2.58801 \dots + πn, x = 1.01722 \dots + πn, x = 2.12437 \dots + πn

+1

度数

x = 31.71747 \dots^{\circ} + 18 0^{\circ} n, x = 148.28252 \dots^{\circ} + 18 0^{\circ} n, x = 58.28252 \dots^{\circ} + 18 0^{\circ} n, x = 121.71747 \dots^{\circ} + 18 0^{\circ} n

求解步骤

cot^{4} (x) - 3 cot^{2} (x) + 1 = 0

用替代法求解

cot^{4} (x) - 3 cot^{2} (x) + 1 = 0

令

: cot (x) = u

u^{4} - 3 u^{2} + 1 = 0

u^{4} - 3 u^{2} + 1 = 0 : u = \frac{3 + 5}{2}, u = - \frac{3 + 5}{2}, u = \frac{3 - 5}{2}, u = - \frac{3 - 5}{2}

u^{4} - 3 u^{2} + 1 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} - 3 v + 1 = 0

解

v^{2} - 3 v + 1 = 0 : v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

v^{2} - 3 v + 1 = 0

使用求根公式求解

v^{2} - 3 v + 1 = 0

二次方程求根公式:

若

a = 1, b = - 3, c = 1

v_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 1}{2 \cdot 1}

(- 3)^{2} - 4 \cdot 1 \cdot 1 = 5

(- 3)^{2} - 4 \cdot 1 \cdot 1

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

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

= 3^{2} - 4 \cdot 1 \cdot 1

数字相乘:

4 \cdot 1 \cdot 1 = 4

= 3^{2} - 4

3^{2} = 9

= 9 - 4

数字相减:

9 - 4 = 5

= 5

v_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 \cdot 1}

将解分隔开

v_{1} = \frac{- ( - 3 ) + 5}{2 \cdot 1}, v_{2} = \frac{- ( - 3 ) - 5}{2 \cdot 1}

v = \frac{- ( - 3 ) + 5}{2 \cdot 1} : \frac{3 + 5}{2}

\frac{- ( - 3 ) + 5}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{3 + 5}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{3 + 5}{2}

v = \frac{- ( - 3 ) - 5}{2 \cdot 1} : \frac{3 - 5}{2}

\frac{- ( - 3 ) - 5}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{3 - 5}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{3 - 5}{2}

二次方程组的解是:

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

代回

v = u^{2}

，求解

u

解

u^{2} = \frac{3 + 5}{2} : u = \frac{3 + 5}{2}, u = - \frac{3 + 5}{2}

u^{2} = \frac{3 + 5}{2}

对于

x^{2} = f (a)

解为

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

u = \frac{3 + 5}{2}, u = - \frac{3 + 5}{2}

解

u^{2} = \frac{3 - 5}{2} : u = \frac{3 - 5}{2}, u = - \frac{3 - 5}{2}

u^{2} = \frac{3 - 5}{2}

对于

x^{2} = f (a)

解为

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

u = \frac{3 - 5}{2}, u = - \frac{3 - 5}{2}

解为

u = \frac{3 + 5}{2}, u = - \frac{3 + 5}{2}, u = \frac{3 - 5}{2}, u = - \frac{3 - 5}{2}

u = cot (x)

代回

cot (x) = \frac{3 + 5}{2}, cot (x) = - \frac{3 + 5}{2}, cot (x) = \frac{3 - 5}{2}, cot (x) = - \frac{3 - 5}{2}

cot (x) = \frac{3 + 5}{2}, cot (x) = - \frac{3 + 5}{2}, cot (x) = \frac{3 - 5}{2}, cot (x) = - \frac{3 - 5}{2}

cot (x) = \frac{3 + 5}{2} : x = arccot \frac{3 + 5}{2} + πn

cot (x) = \frac{3 + 5}{2}

使用反三角函数性质

cot (x) = \frac{3 + 5}{2}

cot (x) = \frac{3 + 5}{2}

的通解

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

x = arccot \frac{3 + 5}{2} + πn

x = arccot \frac{3 + 5}{2} + πn

cot (x) = - \frac{3 + 5}{2} : x = arccot - \frac{3 + 5}{2} + πn

cot (x) = - \frac{3 + 5}{2}

使用反三角函数性质

cot (x) = - \frac{3 + 5}{2}

cot (x) = - \frac{3 + 5}{2}

的通解

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

x = arccot - \frac{3 + 5}{2} + πn

x = arccot - \frac{3 + 5}{2} + πn

cot (x) = \frac{3 - 5}{2} : x = arccot \frac{3 - 5}{2} + πn

cot (x) = \frac{3 - 5}{2}

使用反三角函数性质

cot (x) = \frac{3 - 5}{2}

cot (x) = \frac{3 - 5}{2}

的通解

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

x = arccot \frac{3 - 5}{2} + πn

x = arccot \frac{3 - 5}{2} + πn

cot (x) = - \frac{3 - 5}{2} : x = arccot - \frac{3 - 5}{2} + πn

cot (x) = - \frac{3 - 5}{2}

使用反三角函数性质

cot (x) = - \frac{3 - 5}{2}

cot (x) = - \frac{3 - 5}{2}

的通解

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

x = arccot - \frac{3 - 5}{2} + πn

x = arccot - \frac{3 - 5}{2} + πn

合并所有解

x = arccot \frac{3 + 5}{2} + πn, x = arccot - \frac{3 + 5}{2} + πn, x = arccot \frac{3 - 5}{2} + πn, x = arccot - \frac{3 - 5}{2} + πn

以小数形式表示解

x = 0.55357 \dots + πn, x = 2.58801 \dots + πn, x = 1.01722 \dots + πn, x = 2.12437 \dots + πn

作图

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