zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

csc^2(x)-3csc(x)-2=0

解答

csc^{2} (x) - 3 csc (x) - 2 = 0

解答

x = 0.28460 \dots + 2 πn, x = π - 0.28460 \dots + 2 πn

+1

度数

x = 16.30654 \dots^{\circ} + 36 0^{\circ} n, x = 163.69345 \dots^{\circ} + 36 0^{\circ} n

求解步骤

csc^{2} (x) - 3 csc (x) - 2 = 0

用替代法求解

csc^{2} (x) - 3 csc (x) - 2 = 0

令

: csc (x) = u

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

u^{2} - 3 u - 2 = 0 : u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

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

使用求根公式求解

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

二次方程求根公式:

若

a = 1, b = - 3, c = - 2

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

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

(- 3)^{2} - 4 \cdot 1 \cdot (- 2) = 17

(- 3)^{2} - 4 \cdot 1 \cdot (- 2)

使用法则

- (- a) = a

= (- 3)^{2} + 4 \cdot 1 \cdot 2

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 2

数字相乘:

4 \cdot 1 \cdot 2 = 8

= 3^{2} + 8

3^{2} = 9

= 9 + 8

数字相加:

9 + 8 = 17

= 17

u_{1, 2} = \frac{- ( - 3 ) \pm 17}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 17}{2 \cdot 1}, u_{2} = \frac{- ( - 3 ) - 17}{2 \cdot 1}

u = \frac{- ( - 3 ) + 17}{2 \cdot 1} : \frac{3 + 17}{2}

\frac{- ( - 3 ) + 17}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{3 + 17}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{3 + 17}{2}

u = \frac{- ( - 3 ) - 17}{2 \cdot 1} : \frac{3 - 17}{2}

\frac{- ( - 3 ) - 17}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{3 - 17}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{3 - 17}{2}

二次方程组的解是:

u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

u = csc (x)

代回

csc (x) = \frac{3 + 17}{2}, csc (x) = \frac{3 - 17}{2}

csc (x) = \frac{3 + 17}{2}, csc (x) = \frac{3 - 17}{2}

csc (x) = \frac{3 + 17}{2} : x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

csc (x) = \frac{3 + 17}{2}

使用反三角函数性质

csc (x) = \frac{3 + 17}{2}

csc (x) = \frac{3 + 17}{2}

的通解

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

x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

csc (x) = \frac{3 - 17}{2} :

无解

csc (x) = \frac{3 - 17}{2}

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

无解

合并所有解

x = arccsc (\frac{3 + 17}{2}) + 2 πn, x = π - arccsc (\frac{3 + 17}{2}) + 2 πn

以小数形式表示解

x = 0.28460 \dots + 2 πn, x = π - 0.28460 \dots + 2 πn

作图

流行的例子

3cos(x)=5sin(x)

3 cos (x) = 5 sin (x)

cos(2x-15)=-sin(60-3x)

cos (2 x - 1 5^{\circ}) = - sin (6 0^{\circ} - 3 x)

4sin^2(x)-cos(x)-1=0

4 sin^{2} (x) - cos (x) - 1 = 0

2sin^2(x)+sin(x)-6=0

2 sin^{2} (x) + sin (x) - 6 = 0

tan(θ)= 8/15 ,sin(θ)>0

tan (θ) = \frac{8}{15}, sin (θ) > 0