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受欢迎的三角函数 >

2cot^2(3x)=5csc(3x)-4

解答

2 cot^{2} (3 x) = 5 csc (3 x) - 4

解答

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

+1

度数

x = 1 0^{\circ} + 12 0^{\circ} n, x = 5 0^{\circ} + 12 0^{\circ} n

求解步骤

2 cot^{2} (3 x) = 5 csc (3 x) - 4

两边减去

5 csc (3 x) - 4

2 cot^{2} (3 x) - 5 csc (3 x) + 4 = 0

使用三角恒等式改写

4 + 2 cot^{2} (3 x) - 5 csc (3 x)

使用毕达哥拉斯恒等式:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= 4 + 2 (csc^{2} (3 x) - 1) - 5 csc (3 x)

化简

4 + 2 (csc^{2} (3 x) - 1) - 5 csc (3 x) : 2 csc^{2} (3 x) - 5 csc (3 x) + 2

4 + 2 (csc^{2} (3 x) - 1) - 5 csc (3 x)

乘开

2 (csc^{2} (3 x) - 1) : 2 csc^{2} (3 x) - 2

2 (csc^{2} (3 x) - 1)

使用分配律:

a (b - c) = ab - a c

a = 2, b = csc^{2} (3 x), c = 1

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

数字相乘:

2 \cdot 1 = 2

= 2 csc^{2} (3 x) - 2

= 4 + 2 csc^{2} (3 x) - 2 - 5 csc (3 x)

化简

4 + 2 csc^{2} (3 x) - 2 - 5 csc (3 x) : 2 csc^{2} (3 x) - 5 csc (3 x) + 2

4 + 2 csc^{2} (3 x) - 2 - 5 csc (3 x)

对同类项分组

= 2 csc^{2} (3 x) - 5 csc (3 x) + 4 - 2

数字相加/相减:

4 - 2 = 2

= 2 csc^{2} (3 x) - 5 csc (3 x) + 2

= 2 csc^{2} (3 x) - 5 csc (3 x) + 2

= 2 csc^{2} (3 x) - 5 csc (3 x) + 2

2 + 2 csc^{2} (3 x) - 5 csc (3 x) = 0

用替代法求解

2 + 2 csc^{2} (3 x) - 5 csc (3 x) = 0

令

: csc (3 x) = u

2 + 2 u^{2} - 5 u = 0

2 + 2 u^{2} - 5 u = 0 : u = 2, u = \frac{1}{2}

2 + 2 u^{2} - 5 u = 0

改写成标准形式

a x^{2} + b x + c = 0

2 u^{2} - 5 u + 2 = 0

使用求根公式求解

2 u^{2} - 5 u + 2 = 0

二次方程求根公式:

若

a = 2, b = - 5, c = 2

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

(- 5)^{2} - 4 \cdot 2 \cdot 2 = 3

(- 5)^{2} - 4 \cdot 2 \cdot 2

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 5)^{2} = 5^{2}

= 5^{2} - 4 \cdot 2 \cdot 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

数字相减:

25 - 16 = 9

= 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

u_{1, 2} = \frac{- ( - 5 ) \pm 3}{2 \cdot 2}

将解分隔开

u_{1} = \frac{- ( - 5 ) + 3}{2 \cdot 2}, u_{2} = \frac{- ( - 5 ) - 3}{2 \cdot 2}

u = \frac{- ( - 5 ) + 3}{2 \cdot 2} : 2

\frac{- ( - 5 ) + 3}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{5 + 3}{2 \cdot 2}

数字相加:

5 + 3 = 8

= \frac{8}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= 2

u = \frac{- ( - 5 ) - 3}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 5 ) - 3}{2 \cdot 2}

使用法则

- (- a) = a

= \frac{5 - 3}{2 \cdot 2}

数字相减:

5 - 3 = 2

= \frac{2}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{2}{4}

约分:

2

= \frac{1}{2}

二次方程组的解是:

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

u = csc (3 x)

代回

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

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

csc (3 x) = 2 : x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

csc (3 x) = 2

csc (3 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

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

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

解

3 x = \frac{π}{6} + 2 πn : x = \frac{π}{18} + \frac{2 πn}{3}

3 x = \frac{π}{6} + 2 πn

两边除以

3

3 x = \frac{π}{6} + 2 πn

两边除以

3

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

化简

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

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{\frac{π}{6}}{3} + \frac{2 πn}{3} : \frac{π}{18} + \frac{2 πn}{3}

\frac{\frac{π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{π}{6}}{3} = \frac{π}{18}

\frac{\frac{π}{6}}{3}

使用分式法则:

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

= \frac{π}{6 \cdot 3}

数字相乘:

6 \cdot 3 = 18

= \frac{π}{18}

= \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}

解

3 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{18} + \frac{2 πn}{3}

3 x = \frac{5 π}{6} + 2 πn

两边除以

3

3 x = \frac{5 π}{6} + 2 πn

两边除以

3

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

化简

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

化简

\frac{3 x}{3} : x

\frac{3 x}{3}

数字相除:

\frac{3}{3} = 1

= x

化简

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3} : \frac{5 π}{18} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} + \frac{2 πn}{3}

\frac{\frac{5 π}{6}}{3} = \frac{5 π}{18}

\frac{\frac{5 π}{6}}{3}

使用分式法则:

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

= \frac{5 π}{6 \cdot 3}

数字相乘:

6 \cdot 3 = 18

= \frac{5 π}{18}

= \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{5 π}{18} + \frac{2 πn}{3}

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

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

无解

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

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

无解

合并所有解

x = \frac{π}{18} + \frac{2 πn}{3}, x = \frac{5 π}{18} + \frac{2 πn}{3}

作图

流行的例子

cos (9 x) = \frac{1}{2}

tan(2θ)=-sqrt(3),0<= θ<= 360

tan (2 θ) = - 3, 0^{\circ} \leq θ \leq 36 0^{\circ}

cos(t)-cos(2t)=0

cos (t) - cos (2 t) = 0

cot(2x)=-1,(0,2pi)

cot (2 x) = - 1, (0, 2 π)

sec^2(x)+4sec(x)-12=0

sec^{2} (x) + 4 sec (x) - 12 = 0