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3csc(2x)-4sin(2x)=0

解答

3 csc (2 x) - 4 sin (2 x) = 0

解答

x = \frac{π}{6} + πn, x = \frac{π}{3} + πn, x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

+1

度数

x = 3 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n, x = 12 0^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

求解步骤

3 csc (2 x) - 4 sin (2 x) = 0

使用三角恒等式改写

3 csc (2 x) - 4 sin (2 x)

使用基本三角恒等式:

sin (x) = \frac{1}{csc ( x )}

= 3 csc (2 x) - 4 \cdot \frac{1}{csc ( 2 x )}

4 \cdot \frac{1}{csc ( 2 x )} = \frac{4}{csc ( 2 x )}

4 \cdot \frac{1}{csc ( 2 x )}

分式相乘:

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

= \frac{1 \cdot 4}{csc ( 2 x )}

数字相乘:

1 \cdot 4 = 4

= \frac{4}{csc ( 2 x )}

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

- \frac{4}{csc ( 2 x )} + 3 csc (2 x) = 0

用替代法求解

- \frac{4}{csc ( 2 x )} + 3 csc (2 x) = 0

令

: csc (2 x) = u

- \frac{4}{u} + 3 u = 0

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

- \frac{4}{u} + 3 u = 0

在两边乘以

u

- \frac{4}{u} + 3 u = 0

在两边乘以

u

- \frac{4}{u} u + 3 uu = 0 \cdot u

化简

- \frac{4}{u} u + 3 uu = 0 \cdot u

化简

- \frac{4}{u} u : - 4

- \frac{4}{u} u

分式相乘:

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

= - \frac{4 u}{u}

约分:

u

= - 4

化简

3 uu : 3 u^{2}

3 uu

使用指数法则:

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

uu = u^{1 + 1}

= 3 u^{1 + 1}

数字相加:

1 + 1 = 2

= 3 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

- 4 + 3 u^{2} = 0

- 4 + 3 u^{2} = 0

- 4 + 3 u^{2} = 0

解

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

- 4 + 3 u^{2} = 0

将

4

到右边

- 4 + 3 u^{2} = 0

两边加上

4

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

化简

3 u^{2} = 4

3 u^{2} = 4

两边除以

3

3 u^{2} = 4

两边除以

3

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

化简

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

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

对于

x^{2} = f (a)

解为

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

u = \frac{4}{3}, u = - \frac{4}{3}

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

\frac{4}{3}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \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}

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

- \frac{4}{3}

化简

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

\frac{4}{3}

使用根式运算法则:

假定

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

因式分解数字:

4 = 2^{2}

= 2^{2}

使用根式运算法则:

2^{2} = 2

= 2

= \frac{2}{3}

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

验证解

找到无定义的点（奇点）:

u = 0

取

- \frac{4}{u} + 3 u

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

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

u = csc (2 x)

代回

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

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

csc (2 x) = \frac{2 3}{3} : x = \frac{π}{6} + πn, x = \frac{π}{3} + πn

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

csc (2 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

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

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

解

2 x = \frac{π}{3} + 2 πn : x = \frac{π}{6} + πn

2 x = \frac{π}{3} + 2 πn

两边除以

2

2 x = \frac{π}{3} + 2 πn

两边除以

2

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

化简

\frac{2 x}{2} = \frac{\frac{π}{3}}{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}{3}}{2} + \frac{2 πn}{2} : \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

使用分式法则:

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

= \frac{π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

解

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

2 x = \frac{2 π}{3} + 2 πn

两边除以

2

2 x = \frac{2 π}{3} + 2 πn

两边除以

2

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

化简

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

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

\frac{\frac{2 π}{3}}{2} = \frac{π}{3}

\frac{\frac{2 π}{3}}{2}

使用分式法则:

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

= \frac{2 π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{2 π}{6}

约分:

2

= \frac{π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{3} + πn

x = \frac{π}{6} + πn, x = \frac{π}{3} + πn

csc (2 x) = - \frac{2 3}{3} : x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

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

csc (2 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

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

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

解

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

2 x = \frac{4 π}{3} + 2 πn

两边除以

2

2 x = \frac{4 π}{3} + 2 πn

两边除以

2

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

化简

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

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

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

\frac{\frac{4 π}{3}}{2}

使用分式法则:

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

= \frac{4 π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{4 π}{6}

约分:

2

= \frac{2 π}{3}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

解

2 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{6} + πn

2 x = \frac{5 π}{3} + 2 πn

两边除以

2

2 x = \frac{5 π}{3} + 2 πn

两边除以

2

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

化简

\frac{2 x}{2} = \frac{\frac{5 π}{3}}{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}{3}}{2} + \frac{2 πn}{2} : \frac{5 π}{6} + πn

\frac{\frac{5 π}{3}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{3}}{2} = \frac{5 π}{6}

\frac{\frac{5 π}{3}}{2}

使用分式法则:

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

= \frac{5 π}{3 \cdot 2}

数字相乘:

3 \cdot 2 = 6

= \frac{5 π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

数字相除:

\frac{2}{2} = 1

= πn

= \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

合并所有解

x = \frac{π}{6} + πn, x = \frac{π}{3} + πn, x = \frac{2 π}{3} + πn, x = \frac{5 π}{6} + πn

流行的例子

3tan(B)-4=tan(B)-2

3 tan (B) - 4 = tan (B) - 2

sin(2x)=5cos(2x)

sin (2 x) = 5 cos (2 x)

5 sec (2 x) + 2 = 0

2.1sin(θ)=1.33sin(θ+90)

2.1 sin (θ) = 1.33 sin (θ + 9 0^{\circ})

1sin(45)=1.33sin(θ)

1 sin (4 5^{\circ}) = 1.33 sin (θ)