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受欢迎的三角函数 >

csc^2(θ)+7csc(θ)+12=0

解答

csc^{2} (θ) + 7 csc (θ) + 12 = 0

解答

θ = - 0.33983 \dots + 2 πn, θ = π + 0.33983 \dots + 2 πn, θ = - 0.25268 \dots + 2 πn, θ = π + 0.25268 \dots + 2 πn

+1

度数

θ = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, θ = 199.47122 \dots^{\circ} + 36 0^{\circ} n, θ = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, θ = 194.47751 \dots^{\circ} + 36 0^{\circ} n

求解步骤

csc^{2} (θ) + 7 csc (θ) + 12 = 0

用替代法求解

csc^{2} (θ) + 7 csc (θ) + 12 = 0

令

: csc (θ) = u

u^{2} + 7 u + 12 = 0

u^{2} + 7 u + 12 = 0 : u = - 3, u = - 4

u^{2} + 7 u + 12 = 0

使用求根公式求解

u^{2} + 7 u + 12 = 0

二次方程求根公式:

若

a = 1, b = 7, c = 12

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 1 \cdot 12}{2 \cdot 1}

u_{1, 2} = \frac{- 7 \pm 7 ^{2} - 4 \cdot 1 \cdot 12}{2 \cdot 1}

7^{2} - 4 \cdot 1 \cdot 12 = 1

7^{2} - 4 \cdot 1 \cdot 12

数字相乘:

4 \cdot 1 \cdot 12 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

数字相减:

49 - 48 = 1

= 1

使用法则

1 = 1

= 1

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

将解分隔开

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

u = \frac{- 7 + 1}{2 \cdot 1} : - 3

\frac{- 7 + 1}{2 \cdot 1}

数字相加/相减:

- 7 + 1 = - 6

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 6}{2}

使用分式法则:

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

= - \frac{6}{2}

数字相除:

\frac{6}{2} = 3

= - 3

u = \frac{- 7 - 1}{2 \cdot 1} : - 4

\frac{- 7 - 1}{2 \cdot 1}

数字相减:

- 7 - 1 = - 8

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

数字相乘:

2 \cdot 1 = 2

= \frac{- 8}{2}

使用分式法则:

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

= - \frac{8}{2}

数字相除:

\frac{8}{2} = 4

= - 4

二次方程组的解是:

u = - 3, u = - 4

u = csc (θ)

代回

csc (θ) = - 3, csc (θ) = - 4

csc (θ) = - 3, csc (θ) = - 4

csc (θ) = - 3 : θ = arccsc (- 3) + 2 πn, θ = π + arccsc (3) + 2 πn

csc (θ) = - 3

使用反三角函数性质

csc (θ) = - 3

csc (θ) = - 3

的通解

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

θ = arccsc (- 3) + 2 πn, θ = π + arccsc (3) + 2 πn

θ = arccsc (- 3) + 2 πn, θ = π + arccsc (3) + 2 πn

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

合并所有解

θ = arccsc (- 3) + 2 πn, θ = π + arccsc (3) + 2 πn, θ = arccsc (- 4) + 2 πn, θ = π + arccsc (4) + 2 πn

以小数形式表示解

θ = - 0.33983 \dots + 2 πn, θ = π + 0.33983 \dots + 2 πn, θ = - 0.25268 \dots + 2 πn, θ = π + 0.25268 \dots + 2 πn

作图

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