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受欢迎的三角函数 >

3cot^2(x)-4csc(x)=1

解答

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

解答

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

+1

度数

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

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

两边减去

1

3 cot^{2} (x) - 4 csc (x) - 1 = 0

使用三角恒等式改写

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

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

化简

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

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

乘开

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

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

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

3 \cdot 1 = 3

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

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

化简

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

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

对同类项分组

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

数字相减:

- 1 - 3 = - 4

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

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

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

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

用替代法求解

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

令

: csc (x) = u

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

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

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

改写成标准形式

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

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

使用求根公式求解

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

二次方程求根公式:

若

a = 3, b = - 4, c = - 4

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 3 ( - 4 )}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 3 ( - 4 )}{2 \cdot 3}

(- 4)^{2} - 4 \cdot 3 (- 4) = 8

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

使用法则

- (- a) = a

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

使用指数法则:

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

若

n

是偶数

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

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

数字相乘:

4 \cdot 3 \cdot 4 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

数字相加:

16 + 48 = 64

= 64

因式分解数字:

64 = 8^{2}

= 8^{2}

使用根式运算法则:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 4 ) \pm 8}{2 \cdot 3}

将解分隔开

u_{1} = \frac{- ( - 4 ) + 8}{2 \cdot 3}, u_{2} = \frac{- ( - 4 ) - 8}{2 \cdot 3}

u = \frac{- ( - 4 ) + 8}{2 \cdot 3} : 2

\frac{- ( - 4 ) + 8}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{4 + 8}{2 \cdot 3}

数字相加:

4 + 8 = 12

= \frac{12}{2 \cdot 3}

数字相乘:

2 \cdot 3 = 6

= \frac{12}{6}

数字相除:

\frac{12}{6} = 2

= 2

u = \frac{- ( - 4 ) - 8}{2 \cdot 3} : - \frac{2}{3}

\frac{- ( - 4 ) - 8}{2 \cdot 3}

使用法则

- (- a) = a

= \frac{4 - 8}{2 \cdot 3}

数字相减:

4 - 8 = - 4

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

数字相乘:

2 \cdot 3 = 6

= \frac{- 4}{6}

使用分式法则:

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

= - \frac{4}{6}

约分:

2

= - \frac{2}{3}

二次方程组的解是:

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

u = csc (x)

代回

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

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

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

csc (x) = 2

csc (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

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

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

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

无解

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

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

无解

合并所有解

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

作图

流行的例子

cos(x)=sin(2x)+cos(3x)

cos (x) = sin (2 x) + cos (3 x)

tan(x)sec(x)-2tan(x)=0

tan (x) sec (x) - 2 tan (x) = 0

-9cos^2(θ)+9=17sin(θ)-8

- 9 cos^{2} (θ) + 9 = 17 sin (θ) - 8

4 tan^{2} (θ) - 9 = 0

(sin(90))/(3.37)=(sin(x))/(0.7)

\frac{sin ( 9 0 ^{\circ} )}{3.37} = \frac{sin ( x )}{0.7}