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受欢迎的三角函数 >

-8csc^2(x)+2cot(x)=-23

解答

- 8 csc^{2} (x) + 2 cot (x) = - 23

解答

x = 2.46685 \dots + πn, x = 0.58800 \dots + πn

+1

度数

x = 141.34019 \dots^{\circ} + 18 0^{\circ} n, x = 33.69006 \dots^{\circ} + 18 0^{\circ} n

求解步骤

- 8 csc^{2} (x) + 2 cot (x) = - 23

两边减去

- 23

- 8 csc^{2} (x) + 2 cot (x) + 23 = 0

使用三角恒等式改写

23 + 2 cot (x) - 8 csc^{2} (x)

使用毕达哥拉斯恒等式:

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

= 23 + 2 cot (x) - 8 (1 + cot^{2} (x))

化简

23 + 2 cot (x) - 8 (1 + cot^{2} (x)) : 2 cot (x) - 8 cot^{2} (x) + 15

23 + 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)

使用加减运算法则

+ (- a) = - a

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

数字相乘:

8 \cdot 1 = 8

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

= 23 + 2 cot (x) - 8 - 8 cot^{2} (x)

化简

23 + 2 cot (x) - 8 - 8 cot^{2} (x) : 2 cot (x) - 8 cot^{2} (x) + 15

23 + 2 cot (x) - 8 - 8 cot^{2} (x)

对同类项分组

= 2 cot (x) - 8 cot^{2} (x) + 23 - 8

数字相加/相减:

23 - 8 = 15

= 2 cot (x) - 8 cot^{2} (x) + 15

= 2 cot (x) - 8 cot^{2} (x) + 15

= 2 cot (x) - 8 cot^{2} (x) + 15

15 + 2 cot (x) - 8 cot^{2} (x) = 0

用替代法求解

15 + 2 cot (x) - 8 cot^{2} (x) = 0

令

: cot (x) = u

15 + 2 u - 8 u^{2} = 0

15 + 2 u - 8 u^{2} = 0 : u = - \frac{5}{4}, u = \frac{3}{2}

15 + 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 + 15 = 0

使用求根公式求解

- 8 u^{2} + 2 u + 15 = 0

二次方程求根公式:

若

a = - 8, b = 2, c = 15

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 8 ) \cdot 15}{2 ( - 8 )}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 ( - 8 ) \cdot 15}{2 ( - 8 )}

2^{2} - 4 (- 8) \cdot 15 = 22

2^{2} - 4 (- 8) \cdot 15

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 8 \cdot 15

数字相乘:

4 \cdot 8 \cdot 15 = 480

= 2^{2} + 480

2^{2} = 4

= 4 + 480

数字相加:

4 + 480 = 484

= 484

因式分解数字:

484 = 2 2^{2}

= 2 2^{2}

使用根式运算法则:

n a^{n} = a

2 2^{2} = 22

= 22

u_{1, 2} = \frac{- 2 \pm 22}{2 ( - 8 )}

将解分隔开

u_{1} = \frac{- 2 + 22}{2 ( - 8 )}, u_{2} = \frac{- 2 - 22}{2 ( - 8 )}

u = \frac{- 2 + 22}{2 ( - 8 )} : - \frac{5}{4}

\frac{- 2 + 22}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 2 + 22}{- 2 \cdot 8}

数字相加/相减:

- 2 + 22 = 20

= \frac{20}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{20}{- 16}

使用分式法则:

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

= - \frac{20}{16}

约分:

4

= - \frac{5}{4}

u = \frac{- 2 - 22}{2 ( - 8 )} : \frac{3}{2}

\frac{- 2 - 22}{2 ( - 8 )}

去除括号:

(- a) = - a

= \frac{- 2 - 22}{- 2 \cdot 8}

数字相减:

- 2 - 22 = - 24

= \frac{- 24}{- 2 \cdot 8}

数字相乘:

2 \cdot 8 = 16

= \frac{- 24}{- 16}

使用分式法则:

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

= \frac{24}{16}

约分:

8

= \frac{3}{2}

二次方程组的解是:

u = - \frac{5}{4}, u = \frac{3}{2}

u = cot (x)

代回

cot (x) = - \frac{5}{4}, cot (x) = \frac{3}{2}

cot (x) = - \frac{5}{4}, cot (x) = \frac{3}{2}

cot (x) = - \frac{5}{4} : x = arccot (- \frac{5}{4}) + πn

cot (x) = - \frac{5}{4}

使用反三角函数性质

cot (x) = - \frac{5}{4}

cot (x) = - \frac{5}{4}

的通解

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

x = arccot (- \frac{5}{4}) + πn

x = arccot (- \frac{5}{4}) + πn

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

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

使用反三角函数性质

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

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

的通解

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

x = arccot (\frac{3}{2}) + πn

x = arccot (\frac{3}{2}) + πn

合并所有解

x = arccot (- \frac{5}{4}) + πn, x = arccot (\frac{3}{2}) + πn

以小数形式表示解

x = 2.46685 \dots + πn, x = 0.58800 \dots + πn

作图

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