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受欢迎的三角函数 >

csc^4(2x)-4=0

解答

csc^{4} (2 x) - 4 = 0

解答

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn, x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

+1

度数

x = 22. 5^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n, x = 112. 5^{\circ} + 18 0^{\circ} n, x = 157. 5^{\circ} + 18 0^{\circ} n

求解步骤

csc^{4} (2 x) - 4 = 0

用替代法求解

csc^{4} (2 x) - 4 = 0

令

: csc (2 x) = u

u^{4} - 4 = 0

u^{4} - 4 = 0 : u = 2, u = - 2, u = 2 i, u = - 2 i

u^{4} - 4 = 0

将

4

到右边

u^{4} - 4 = 0

两边加上

4

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

化简

u^{4} = 4

u^{4} = 4

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

v^{2} = 4

解

v^{2} = 4 : v = 4, v = - 4

v^{2} = 4

对于

(g (x))^{2} = f (a)

解为

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

v = 4, v = - 4

v = 4, v = - 4

代回

v = u^{2}

，求解

u

解

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

化简

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

对于

x^{2} = f (a)

解为

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

u = 2, u = - 2

解

u^{2} = - 4 : u = 2 i, u = - 2 i

u^{2} = - 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

u^{2} = - 2

对于

x^{2} = f (a)

解为

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

u = - 2, u = - - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

化简

- - 2 : - 2 i

- - 2

化简

- 2 : 2 i

- 2

使用根式运算法则:

- a = - 1 a

- 2 = - 1 2

= - 1 2

使用虚数运算法则:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

解为

u = 2, u = - 2, u = 2 i, u = - 2 i

u = csc (2 x)

代回

csc (2 x) = 2, csc (2 x) = - 2, csc (2 x) = 2 i, csc (2 x) = - 2 i

csc (2 x) = 2, csc (2 x) = - 2, csc (2 x) = 2 i, csc (2 x) = - 2 i

csc (2 x) = 2 : x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

csc (2 x) = 2

csc (2 x) = 2

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

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

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

解

2 x = \frac{π}{4} + 2 πn : x = \frac{π}{8} + πn

2 x = \frac{π}{4} + 2 πn

两边除以

2

2 x = \frac{π}{4} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

解

2 x = \frac{3 π}{4} + 2 πn : x = \frac{3 π}{8} + πn

2 x = \frac{3 π}{4} + 2 πn

两边除以

2

2 x = \frac{3 π}{4} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2} : \frac{3 π}{8} + πn

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{3 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{3 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

csc (2 x) = - 2 : x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

csc (2 x) = - 2

csc (2 x) = - 2

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

2 x = \frac{5 π}{4} + 2 πn, 2 x = \frac{7 π}{4} + 2 πn

2 x = \frac{5 π}{4} + 2 πn, 2 x = \frac{7 π}{4} + 2 πn

解

2 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{8} + πn

2 x = \frac{5 π}{4} + 2 πn

两边除以

2

2 x = \frac{5 π}{4} + 2 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2} : \frac{5 π}{8} + πn

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{4}}{2} = \frac{5 π}{8}

\frac{\frac{5 π}{4}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{5 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

解

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{8} + πn

2 x = \frac{7 π}{4} + 2 πn

两边除以

2

2 x = \frac{7 π}{4} + 2 πn

两边除以

2

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π}{8} + πn

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{4}}{2} = \frac{7 π}{8}

\frac{\frac{7 π}{4}}{2}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{7 π}{4 \cdot 2}

数字相乘:

4 \cdot 2 = 8

= \frac{7 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

csc (2 x) = 2 i :

无解

csc (2 x) = 2 i

无解

csc (2 x) = - 2 i :

无解

csc (2 x) = - 2 i

无解

合并所有解

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn, x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

作图

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