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受欢迎的三角函数 >

8csc^2(x)-2cot(x)=9

解答

8 csc^{2} (x) - 2 cot (x) = 9

解答

x = 1.10714 \dots + πn, x = 1.81577 \dots + πn

+1

度数

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = 104.03624 \dots^{\circ} + 18 0^{\circ} n

求解步骤

8 csc^{2} (x) - 2 cot (x) = 9

两边减去

9

8 csc^{2} (x) - 2 cot (x) - 9 = 0

使用三角恒等式改写

- 9 - 2 cot (x) + 8 csc^{2} (x)

使用毕达哥拉斯恒等式:

csc^{2} (x) = 1 + cot^{2} (x)

= - 9 - 2 cot (x) + 8 (1 + cot^{2} (x))

化简

- 9 - 2 cot (x) + 8 (1 + cot^{2} (x)) : 8 cot^{2} (x) - 2 cot (x) - 1

- 9 - 2 cot (x) + 8 (1 + cot^{2} (x))

乘开

8 (1 + cot^{2} (x)) : 8 + 8 cot^{2} (x)

8 (1 + cot^{2} (x))

使用分配律:

a (b + c) = ab + a c

a = 8, b = 1, c = cot^{2} (x)

= 8 \cdot 1 + 8 cot^{2} (x)

数字相乘:

8 \cdot 1 = 8

= 8 + 8 cot^{2} (x)

= - 9 - 2 cot (x) + 8 + 8 cot^{2} (x)

化简

- 9 - 2 cot (x) + 8 + 8 cot^{2} (x) : 8 cot^{2} (x) - 2 cot (x) - 1

- 9 - 2 cot (x) + 8 + 8 cot^{2} (x)

对同类项分组

= - 2 cot (x) + 8 cot^{2} (x) - 9 + 8

数字相加/相减:

- 9 + 8 = - 1

= 8 cot^{2} (x) - 2 cot (x) - 1

= 8 cot^{2} (x) - 2 cot (x) - 1

= 8 cot^{2} (x) - 2 cot (x) - 1

- 1 - 2 cot (x) + 8 cot^{2} (x) = 0

用替代法求解

- 1 - 2 cot (x) + 8 cot^{2} (x) = 0

令

: cot (x) = u

- 1 - 2 u + 8 u^{2} = 0

- 1 - 2 u + 8 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{4}

- 1 - 2 u + 8 u^{2} = 0

改写成标准形式

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

8 u^{2} - 2 u - 1 = 0

使用求根公式求解

8 u^{2} - 2 u - 1 = 0

二次方程求根公式:

若

a = 8, b = - 2, c = - 1

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 8 ( - 1 )}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 \cdot 8 ( - 1 )}{2 \cdot 8}

(- 2)^{2} - 4 \cdot 8 (- 1) = 6

(- 2)^{2} - 4 \cdot 8 (- 1)

使用法则

- (- a) = a

= (- 2)^{2} + 4 \cdot 8 \cdot 1

使用指数法则:

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

若

n

是偶数

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

= 2^{2} + 4 \cdot 8 \cdot 1

数字相乘:

4 \cdot 8 \cdot 1 = 32

= 2^{2} + 32

2^{2} = 4

= 4 + 32

数字相加:

4 + 32 = 36

= 36

因式分解数字:

36 = 6^{2}

= 6^{2}

使用根式运算法则:

n a^{n} = a

6^{2} = 6

= 6

u_{1, 2} = \frac{- ( - 2 ) \pm 6}{2 \cdot 8}

将解分隔开

u_{1} = \frac{- ( - 2 ) + 6}{2 \cdot 8}, u_{2} = \frac{- ( - 2 ) - 6}{2 \cdot 8}

u = \frac{- ( - 2 ) + 6}{2 \cdot 8} : \frac{1}{2}

\frac{- ( - 2 ) + 6}{2 \cdot 8}

使用法则

- (- a) = a

= \frac{2 + 6}{2 \cdot 8}

数字相加:

2 + 6 = 8

= \frac{8}{2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{8}{16}

约分:

8

= \frac{1}{2}

u = \frac{- ( - 2 ) - 6}{2 \cdot 8} : - \frac{1}{4}

\frac{- ( - 2 ) - 6}{2 \cdot 8}

使用法则

- (- a) = a

= \frac{2 - 6}{2 \cdot 8}

数字相减:

2 - 6 = - 4

= \frac{- 4}{2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 4}{16}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{16}

约分:

4

= - \frac{1}{4}

二次方程组的解是:

u = \frac{1}{2}, u = - \frac{1}{4}

u = cot (x)

代回

cot (x) = \frac{1}{2}, cot (x) = - \frac{1}{4}

cot (x) = \frac{1}{2}, cot (x) = - \frac{1}{4}

cot (x) = \frac{1}{2} : x = arccot (\frac{1}{2}) + πn

cot (x) = \frac{1}{2}

使用反三角函数性质

cot (x) = \frac{1}{2}

cot (x) = \frac{1}{2}

的通解

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

x = arccot (\frac{1}{2}) + πn

x = arccot (\frac{1}{2}) + πn

cot (x) = - \frac{1}{4} : x = arccot (- \frac{1}{4}) + πn

cot (x) = - \frac{1}{4}

使用反三角函数性质

cot (x) = - \frac{1}{4}

cot (x) = - \frac{1}{4}

的通解

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

x = arccot (- \frac{1}{4}) + πn

x = arccot (- \frac{1}{4}) + πn

合并所有解

x = arccot (\frac{1}{2}) + πn, x = arccot (- \frac{1}{4}) + πn

以小数形式表示解

x = 1.10714 \dots + πn, x = 1.81577 \dots + πn

作图

流行的例子

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