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受欢迎的三角函数 >

csc^2(θ)+2csc(θ)-24=0

解答

csc^{2} (θ) + 2 csc (θ) - 24 = 0

解答

θ = 0.25268 \dots + 2 πn, θ = π - 0.25268 \dots + 2 πn, θ = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn

+1

度数

θ = 14.47751 \dots^{\circ} + 36 0^{\circ} n, θ = 165.52248 \dots^{\circ} + 36 0^{\circ} n, θ = - 9.59406 \dots^{\circ} + 36 0^{\circ} n, θ = 189.59406 \dots^{\circ} + 36 0^{\circ} n

求解步骤

csc^{2} (θ) + 2 csc (θ) - 24 = 0

用替代法求解

csc^{2} (θ) + 2 csc (θ) - 24 = 0

令

: csc (θ) = u

u^{2} + 2 u - 24 = 0

u^{2} + 2 u - 24 = 0 : u = 4, u = - 6

u^{2} + 2 u - 24 = 0

使用求根公式求解

u^{2} + 2 u - 24 = 0

二次方程求根公式:

若

a = 1, b = 2, c = - 24

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 24 )}{2 \cdot 1}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 24 )}{2 \cdot 1}

2^{2} - 4 \cdot 1 \cdot (- 24) = 10

2^{2} - 4 \cdot 1 \cdot (- 24)

使用法则

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 24

数字相乘:

4 \cdot 1 \cdot 24 = 96

= 2^{2} + 96

2^{2} = 4

= 4 + 96

数字相加:

4 + 96 = 100

= 100

因式分解数字:

100 = 1 0^{2}

= 1 0^{2}

使用根式运算法则:

n a^{n} = a

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 2 \pm 10}{2 \cdot 1}

将解分隔开

u_{1} = \frac{- 2 + 10}{2 \cdot 1}, u_{2} = \frac{- 2 - 10}{2 \cdot 1}

u = \frac{- 2 + 10}{2 \cdot 1} : 4

\frac{- 2 + 10}{2 \cdot 1}

数字相加/相减:

- 2 + 10 = 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 - 10}{2 \cdot 1} : - 6

\frac{- 2 - 10}{2 \cdot 1}

数字相减:

- 2 - 10 = - 12

= \frac{- 12}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 12}{2}

使用分式法则:

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

= - \frac{12}{2}

数字相除:

\frac{12}{2} = 6

= - 6

二次方程组的解是:

u = 4, u = - 6

u = csc (θ)

代回

csc (θ) = 4, csc (θ) = - 6

csc (θ) = 4, csc (θ) = - 6

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) = - 6 : θ = arccsc (- 6) + 2 πn, θ = π + arccsc (6) + 2 πn

csc (θ) = - 6

使用反三角函数性质

csc (θ) = - 6

csc (θ) = - 6

的通解

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

θ = arccsc (- 6) + 2 πn, θ = π + arccsc (6) + 2 πn

θ = arccsc (- 6) + 2 πn, θ = π + arccsc (6) + 2 πn

合并所有解

θ = arccsc (4) + 2 πn, θ = π - arccsc (4) + 2 πn, θ = arccsc (- 6) + 2 πn, θ = π + arccsc (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 = - 0.16744 \dots + 2 πn, θ = π + 0.16744 \dots + 2 πn

作图

流行的例子

2sin(θ)=2cos(2θ)

2 sin (θ) = 2 cos (2 θ)

2 cos^{2} (θ) = 1

sec(θ)=-(2sqrt(3))/3

sec (θ) = - \frac{2 3}{3}

cos (θ) = \frac{8}{17}

2tan(x/2)-csc(x)=0

2 tan (\frac{x}{2}) - csc (x) = 0