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受欢迎的三角函数 >

csc^2(θ)-2csc(θ)-8=0

解答

csc^{2} (θ) - 2 csc (θ) - 8 = 0

解答

θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

+1

度数

θ = 14.47751 \dots^{\circ} + 36 0^{\circ} n, θ = 165.52248 \dots^{\circ} + 36 0^{\circ} n, θ = 21 0^{\circ} + 36 0^{\circ} n, θ = 33 0^{\circ} + 36 0^{\circ} n

求解步骤

csc^{2} (θ) - 2 csc (θ) - 8 = 0

用替代法求解

csc^{2} (θ) - 2 csc (θ) - 8 = 0

令

: csc (θ) = u

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

u^{2} - 2 u - 8 = 0 : u = 4, u = - 2

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

使用求根公式求解

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

二次方程求根公式:

若

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

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

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

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

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 1 \cdot 8 = 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 1}

将解分隔开

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

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

\frac{- ( - 2 ) + 6}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{2 + 6}{2 \cdot 1}

数字相加:

2 + 6 = 8

= \frac{8}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{8}{2}

数字相除:

\frac{8}{2} = 4

= 4

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

\frac{- ( - 2 ) - 6}{2 \cdot 1}

使用法则

- (- a) = a

= \frac{2 - 6}{2 \cdot 1}

数字相减:

2 - 6 = - 4

= \frac{- 4}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 4}{2}

使用分式法则:

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

= - \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= - 2

二次方程组的解是:

u = 4, u = - 2

u = csc (θ)

代回

csc (θ) = 4, csc (θ) = - 2

csc (θ) = 4, csc (θ) = - 2

csc (θ) = 4 : θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn

csc (θ) = 4

使用反三角函数性质

csc (θ) = 4

csc (θ) = 4

的通解

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

θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn

θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn

csc (θ) = - 2 : θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

csc (θ) = - 2

csc (θ) = - 2

的通解

csc (x)

周期表（周期为

2 πn

）：

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

合并所有解

θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

以小数形式表示解

θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

作图

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