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受欢迎的三角函数 >

csc^3(x)+csc^2(x)= 4/3 csc(x)+4/3

解答

csc^{3} (x) + csc^{2} (x) = \frac{4}{3} csc (x) + \frac{4}{3}

解答

x = \frac{3 π}{2} + 2 πn, 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 = 27 0^{\circ} + 36 0^{\circ} n, 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

csc^{3} (x) + csc^{2} (x) = \frac{4}{3} csc (x) + \frac{4}{3}

用替代法求解

csc^{3} (x) + csc^{2} (x) = \frac{4}{3} csc (x) + \frac{4}{3}

令

: csc (x) = u

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3}

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3} : u = - 1, u = - \frac{2 3}{3}, u = \frac{2 3}{3}

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3}

在两边乘以

3

u^{3} + u^{2} = \frac{4}{3} u + \frac{4}{3}

在两边乘以

3

u^{3} \cdot 3 + u^{2} \cdot 3 = \frac{4}{3} u \cdot 3 + \frac{4}{3} \cdot 3

化简

3 u^{3} + 3 u^{2} = 4 u + 4

3 u^{3} + 3 u^{2} = 4 u + 4

将

4

para o lado esquerdo

3 u^{3} + 3 u^{2} = 4 u + 4

两边减去

4

3 u^{3} + 3 u^{2} - 4 = 4 u + 4 - 4

化简

3 u^{3} + 3 u^{2} - 4 = 4 u

3 u^{3} + 3 u^{2} - 4 = 4 u

将

4 u

para o lado esquerdo

3 u^{3} + 3 u^{2} - 4 = 4 u

两边减去

4 u

3 u^{3} + 3 u^{2} - 4 - 4 u = 4 u - 4 u

化简

3 u^{3} + 3 u^{2} - 4 - 4 u = 0

3 u^{3} + 3 u^{2} - 4 - 4 u = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

3 u^{3} + 3 u^{2} - 4 u - 4 = 0

因式分解

3 u^{3} + 3 u^{2} - 4 u - 4 : (u + 1) (3 u + 2) (3 u - 2)

3 u^{3} + 3 u^{2} - 4 u - 4

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

从

- 4 u - 4

分解出因式

- 4

:

- 4 (u + 1)

- 4 u - 4

因式分解出通项

- 4

= - 4 (u + 1)

从

3 u^{3} + 3 u^{2}

分解出因式

3 u^{2}

:

3 u^{2} (u + 1)

3 u^{3} + 3 u^{2}

使用指数法则:

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

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

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

因式分解出通项

3 u^{2}

= 3 u^{2} (u + 1)

= - 4 (u + 1) + 3 u^{2} (u + 1)

因式分解出通项

u + 1

= (u + 1) (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 + 1) (3 u + 2) (3 u - 2)

(u + 1) (3 u + 2) (3 u - 2) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

u + 1 = 0 or 3 u + 2 = 0 or 3 u - 2 = 0

解

u + 1 = 0 : u = - 1

u + 1 = 0

将

1

到右边

u + 1 = 0

两边减去

1

u + 1 - 1 = 0 - 1

化简

u = - 1

u = - 1

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 = - 1, u = - \frac{2 3}{3}, u = \frac{2 3}{3}

u = csc (x)

代回

csc (x) = - 1, csc (x) = - \frac{2 3}{3}, csc (x) = \frac{2 3}{3}

csc (x) = - 1, csc (x) = - \frac{2 3}{3}, csc (x) = \frac{2 3}{3}

csc (x) = - 1 : x = \frac{3 π}{2} + 2 πn

csc (x) = - 1

csc (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

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{3 π}{2} + 2 πn, 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

流行的例子

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