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受欢迎的三角函数 >

2csc^2(x)-2csc(x)-1=0

解答

2 csc^{2} (x) - 2 csc (x) - 1 = 0

解答

x = 0.82132 \dots + 2 πn, x = π - 0.82132 \dots + 2 πn

+1

度数

x = 47.05859 \dots^{\circ} + 36 0^{\circ} n, x = 132.94140 \dots^{\circ} + 36 0^{\circ} n

求解步骤

2 csc^{2} (x) - 2 csc (x) - 1 = 0

用替代法求解

2 csc^{2} (x) - 2 csc (x) - 1 = 0

令

: csc (x) = u

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

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

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

(- 2)^{2} - 4 \cdot 2 (- 1) = 23

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

使用法则

- (- a) = a

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

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 2^{2} + 8

2^{2} = 4

= 4 + 8

数字相加:

4 + 8 = 12

= 12

12

质因数分解:

2^{2} \cdot 3

12

12

除以

212 = 6 \cdot 2

= 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

使用根式运算法则:

n ab = n a n b

= 3 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 23

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

将解分隔开

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

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

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

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 2 = 4

= \frac{2 + 2 3}{4}

分解

2 + 23 : 2 (1 + 3)

2 + 23

改写为

= 2 \cdot 1 + 23

因式分解出通项

2

= 2 (1 + 3)

= \frac{2 ( 1 + 3 )}{4}

约分:

2

= \frac{1 + 3}{2}

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

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

使用法则

- (- a) = a

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

数字相乘:

2 \cdot 2 = 4

= \frac{2 - 2 3}{4}

分解

2 - 23 : 2 (1 - 3)

2 - 23

改写为

= 2 \cdot 1 - 23

因式分解出通项

2

= 2 (1 - 3)

= \frac{2 ( 1 - 3 )}{4}

约分:

2

= \frac{1 - 3}{2}

二次方程组的解是:

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

u = csc (x)

代回

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

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

csc (x) = \frac{1 + 3}{2} : x = arccsc (\frac{1 + 3}{2}) + 2 πn, x = π - arccsc (\frac{1 + 3}{2}) + 2 πn

csc (x) = \frac{1 + 3}{2}

使用反三角函数性质

csc (x) = \frac{1 + 3}{2}

csc (x) = \frac{1 + 3}{2}

的通解

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

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

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

csc (x) = \frac{1 - 3}{2} :

无解

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

csc (x) \leq - 1 or csc (x) \geq 1

无解

合并所有解

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

以小数形式表示解

x = 0.82132 \dots + 2 πn, x = π - 0.82132 \dots + 2 πn

作图

流行的例子

sin(x)-cos(x)= 1/(sqrt(2))

sin (x) - cos (x) = \frac{1}{2}

cos (x) = \frac{6}{15}

sin (θ) = \frac{7}{12}

3cos(2x)+2sin^2(x)=2

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

solvefor x,cos(3x)=1

so l v e f or x, cos (3 x) = 1