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受欢迎的三角函数 >

2csc^2(x)=5-5cot(x)

解答

2 csc^{2} (x) = 5 - 5 cot (x)

解答

x = 1.10714 \dots + πn, x = 2.81984 \dots + πn

+1

度数

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = 161.56505 \dots^{\circ} + 18 0^{\circ} n

求解步骤

2 csc^{2} (x) = 5 - 5 cot (x)

两边减去

5 - 5 cot (x)

2 csc^{2} (x) - 5 + 5 cot (x) = 0

使用三角恒等式改写

- 5 + 2 csc^{2} (x) + 5 cot (x)

使用毕达哥拉斯恒等式:

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

= - 5 + 2 (1 + cot^{2} (x)) + 5 cot (x)

化简

- 5 + 2 (1 + cot^{2} (x)) + 5 cot (x) : 2 cot^{2} (x) + 5 cot (x) - 3

- 5 + 2 (1 + cot^{2} (x)) + 5 cot (x)

乘开

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

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

使用分配律:

a (b + c) = ab + a c

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

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

数字相乘:

2 \cdot 1 = 2

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

= - 5 + 2 + 2 cot^{2} (x) + 5 cot (x)

数字相加/相减:

- 5 + 2 = - 3

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

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

- 3 + 2 cot^{2} (x) + 5 cot (x) = 0

用替代法求解

- 3 + 2 cot^{2} (x) + 5 cot (x) = 0

令

: cot (x) = u

- 3 + 2 u^{2} + 5 u = 0

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

- 3 + 2 u^{2} + 5 u = 0

改写成标准形式

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

2 u^{2} + 5 u - 3 = 0

使用求根公式求解

2 u^{2} + 5 u - 3 = 0

二次方程求根公式:

若

a = 2, b = 5, c = - 3

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

5^{2} - 4 \cdot 2 (- 3) = 7

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

使用法则

- (- a) = a

= 5^{2} + 4 \cdot 2 \cdot 3

数字相乘:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

数字相加:

25 + 24 = 49

= 49

因式分解数字:

49 = 7^{2}

= 7^{2}

使用根式运算法则:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 5 \pm 7}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- 5 + 7}{2 \cdot 2}, u_{2} = \frac{- 5 - 7}{2 \cdot 2}

u = \frac{- 5 + 7}{2 \cdot 2} : \frac{1}{2}

\frac{- 5 + 7}{2 \cdot 2}

数字相加/相减:

- 5 + 7 = 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

约分:

2

= \frac{1}{2}

u = \frac{- 5 - 7}{2 \cdot 2} : - 3

\frac{- 5 - 7}{2 \cdot 2}

数字相减:

- 5 - 7 = - 12

= \frac{- 12}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 12}{4}

使用分式法则:

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

= - \frac{12}{4}

数字相除:

\frac{12}{4} = 3

= - 3

二次方程组的解是:

u = \frac{1}{2}, u = - 3

u = cot (x)

代回

cot (x) = \frac{1}{2}, cot (x) = - 3

cot (x) = \frac{1}{2}, cot (x) = - 3

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) = - 3 : x = arccot (- 3) + πn

cot (x) = - 3

使用反三角函数性质

cot (x) = - 3

cot (x) = - 3

的通解

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

x = arccot (- 3) + πn

x = arccot (- 3) + πn

合并所有解

x = arccot (\frac{1}{2}) + πn, x = arccot (- 3) + πn

以小数形式表示解

x = 1.10714 \dots + πn, x = 2.81984 \dots + πn

作图

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2 sin^{2} (x) = 3 sin (x)

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