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受欢迎的三角函数 >

3cot(x)-1=cot^2(x)

解答

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

解答

x = 0.36486 \dots + πn, x = 1.20593 \dots + πn

+1

度数

x = 20.90515 \dots^{\circ} + 18 0^{\circ} n, x = 69.09484 \dots^{\circ} + 18 0^{\circ} n

求解步骤

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

用替代法求解

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

令

: cot (x) = u

3 u - 1 = u^{2}

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

3 u - 1 = u^{2}

交换两边

u^{2} = 3 u - 1

将

1

para o lado esquerdo

u^{2} = 3 u - 1

两边加上

1

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

化简

u^{2} + 1 = 3 u

u^{2} + 1 = 3 u

将

3 u

para o lado esquerdo

u^{2} + 1 = 3 u

两边减去

3 u

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

化简

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

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

改写成标准形式

a x^{2} + b x + c = 0

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

u_{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

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

将解分隔开

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

u = \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}

u = \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:

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} : 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

以小数形式表示解

x = 0.36486 \dots + πn, x = 1.20593 \dots + πn

作图

流行的例子

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cos (t) = \frac{15}{17}

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sec^2(x)-6sec(x)=16

sec^{2} (x) - 6 sec (x) = 16