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受欢迎的三角函数 >

4csc^2(θ)-9=0

解答

4 csc^{2} (θ) - 9 = 0

解答

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

+1

度数

θ = 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 138.18968 \dots^{\circ} + 36 0^{\circ} n, θ = - 41.81031 \dots^{\circ} + 36 0^{\circ} n, θ = 221.81031 \dots^{\circ} + 36 0^{\circ} n

求解步骤

4 csc^{2} (θ) - 9 = 0

用替代法求解

4 csc^{2} (θ) - 9 = 0

令

: csc (θ) = u

4 u^{2} - 9 = 0

4 u^{2} - 9 = 0 : u = \frac{3}{2}, u = - \frac{3}{2}

4 u^{2} - 9 = 0

将

9

到右边

4 u^{2} - 9 = 0

两边加上

9

4 u^{2} - 9 + 9 = 0 + 9

化简

4 u^{2} = 9

4 u^{2} = 9

两边除以

4

4 u^{2} = 9

两边除以

4

\frac{4 u ^{2}}{4} = \frac{9}{4}

化简

u^{2} = \frac{9}{4}

u^{2} = \frac{9}{4}

对于

x^{2} = f (a)

解为

x = f (a), - f (a)

u = \frac{9}{4}, u = - \frac{9}{4}

\frac{9}{4} = \frac{3}{2}

\frac{9}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{9}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{9}{2}

9 = 3

9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{2}

- \frac{9}{4} = - \frac{3}{2}

- \frac{9}{4}

化简

\frac{9}{4} : \frac{3}{2}

\frac{9}{4}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b},

假定

a \geq 0, b \geq 0

= \frac{9}{4}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 2

= \frac{9}{2}

9 = 3

9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

n a^{n} = a

3^{2} = 3

= 3

= \frac{3}{2}

= - \frac{3}{2}

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

u = csc (θ)

代回

csc (θ) = \frac{3}{2}, csc (θ) = - \frac{3}{2}

csc (θ) = \frac{3}{2}, csc (θ) = - \frac{3}{2}

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

csc (θ) = \frac{3}{2}

使用反三角函数性质

csc (θ) = \frac{3}{2}

csc (θ) = \frac{3}{2}

的通解

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

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

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

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

csc (θ) = - \frac{3}{2}

使用反三角函数性质

csc (θ) = - \frac{3}{2}

csc (θ) = - \frac{3}{2}

的通解

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

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

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

合并所有解

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

以小数形式表示解

θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn, θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

作图

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