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受欢迎的三角函数 >

3csc^3(x)-4csc(x)=0

解答

3 csc^{3} (x) - 4 csc (x) = 0

解答

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

+1

度数

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

求解步骤

3 csc^{3} (x) - 4 csc (x) = 0

用替代法求解

3 csc^{3} (x) - 4 csc (x) = 0

令

: csc (x) = u

3 u^{3} - 4 u = 0

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

3 u^{3} - 4 u = 0

因式分解

3 u^{3} - 4 u : u (3 u + 2) (3 u - 2)

3 u^{3} - 4 u

因式分解出通项

u : u (3 u^{2} - 4)

3 u^{3} - 4 u

使用指数法则:

a^{b + c} = a^{b} a^{c}

u^{3} = u^{2} u

= 3 u^{2} u - 4 u

因式分解出通项

u

= u (3 u^{2} - 4)

= u (3 u^{2} - 4)

分解

3 u^{2} - 4 : (3 u + 2) (3 u - 2)

3 u^{2} - 4

将

3 u^{2} - 4

改写为

(3 u)^{2} - 2^{2}

3 u^{2} - 4

使用根式运算法则:

a = (a)^{2}

3 = (3)^{2}

= (3)^{2} u^{2} - 4

将

4

改写为

2^{2}

= (3)^{2} u^{2} - 2^{2}

使用指数法则:

a^{m} b^{m} = (ab)^{m}

(3)^{2} u^{2} = (3 u)^{2}

= (3 u)^{2} - 2^{2}

= (3 u)^{2} - 2^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(3 u)^{2} - 2^{2} = (3 u + 2) (3 u - 2)

= (3 u + 2) (3 u - 2)

= u (3 u + 2) (3 u - 2)

u (3 u + 2) (3 u - 2) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u = 0 or 3 u + 2 = 0 or 3 u - 2 = 0

解

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

3 u + 2 = 0

将

2

到右边

3 u + 2 = 0

两边减去

2

3 u + 2 - 2 = 0 - 2

化简

3 u = - 2

3 u = - 2

两边除以

3

3 u = - 2

两边除以

3

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

化简

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

化简

\frac{3 u}{3} : u

\frac{3 u}{3}

约分:

3

= u

化简

\frac{- 2}{3} : - \frac{2 3}{3}

\frac{- 2}{3}

使用分式法则:

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

= - \frac{2}{3}

- \frac{2}{3}

有理化:

- \frac{2 3}{3}

- \frac{2}{3}

乘以共轭根式

\frac{3}{3}

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

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

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

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

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

解

3 u - 2 = 0 : u = \frac{2 3}{3}

3 u - 2 = 0

将

2

到右边

3 u - 2 = 0

两边加上

2

3 u - 2 + 2 = 0 + 2

化简

3 u = 2

3 u = 2

两边除以

3

3 u = 2

两边除以

3

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

化简

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

化简

\frac{3 u}{3} : u

\frac{3 u}{3}

约分:

3

= u

化简

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

乘以共轭根式

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

u = \frac{2 3}{3}

u = \frac{2 3}{3}

u = \frac{2 3}{3}

解为

u = 0, u = - \frac{2 3}{3}, u = \frac{2 3}{3}

u = csc (x)

代回

csc (x) = 0, csc (x) = - \frac{2 3}{3}, csc (x) = \frac{2 3}{3}

csc (x) = 0, csc (x) = - \frac{2 3}{3}, csc (x) = \frac{2 3}{3}

csc (x) = 0 :

无解

csc (x) = 0

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

无解

csc (x) = - \frac{2 3}{3} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

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

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

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

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

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

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

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

合并所有解

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

作图

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