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受欢迎的三角函数 >

4csc(x)+sin(x)+5=0

解答

4 csc (x) + sin (x) + 5 = 0

解答

x = \frac{3 π}{2} + 2 πn

+1

度数

x = 27 0^{\circ} + 36 0^{\circ} n

求解步骤

4 csc (x) + sin (x) + 5 = 0

使用三角恒等式改写

5 + sin (x) + 4 csc (x)

使用基本三角恒等式:

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

= 5 + \frac{1}{csc ( x )} + 4 csc (x)

5 + \frac{1}{csc ( x )} + 4 csc (x) = 0

用替代法求解

5 + \frac{1}{csc ( x )} + 4 csc (x) = 0

令

: csc (x) = u

5 + \frac{1}{u} + 4 u = 0

5 + \frac{1}{u} + 4 u = 0 : u = - \frac{1}{4}, u = - 1

5 + \frac{1}{u} + 4 u = 0

在两边乘以

u

5 + \frac{1}{u} + 4 u = 0

在两边乘以

u

5 u + \frac{1}{u} u + 4 uu = 0 \cdot u

化简

5 u + \frac{1}{u} u + 4 uu = 0 \cdot u

化简

\frac{1}{u} u : 1

\frac{1}{u} u

分式相乘:

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

= \frac{1 \cdot u}{u}

约分:

u

= 1

化简

4 uu : 4 u^{2}

4 uu

使用指数法则:

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

uu = u^{1 + 1}

= 4 u^{1 + 1}

数字相加:

1 + 1 = 2

= 4 u^{2}

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

5 u + 1 + 4 u^{2} = 0

5 u + 1 + 4 u^{2} = 0

5 u + 1 + 4 u^{2} = 0

解

5 u + 1 + 4 u^{2} = 0 : u = - \frac{1}{4}, u = - 1

5 u + 1 + 4 u^{2} = 0

改写成标准形式

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

4 u^{2} + 5 u + 1 = 0

使用求根公式求解

4 u^{2} + 5 u + 1 = 0

二次方程求根公式:

若

a = 4, b = 5, c = 1

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

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

5^{2} - 4 \cdot 4 \cdot 1 = 3

5^{2} - 4 \cdot 4 \cdot 1

数字相乘:

4 \cdot 4 \cdot 1 = 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:

3^{2} = 3

= 3

u_{1, 2} = \frac{- 5 \pm 3}{2 \cdot 4}

将解分隔开

u_{1} = \frac{- 5 + 3}{2 \cdot 4}, u_{2} = \frac{- 5 - 3}{2 \cdot 4}

u = \frac{- 5 + 3}{2 \cdot 4} : - \frac{1}{4}

\frac{- 5 + 3}{2 \cdot 4}

数字相加/相减:

- 5 + 3 = - 2

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 2}{8}

使用分式法则:

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

= - \frac{2}{8}

约分:

2

= - \frac{1}{4}

u = \frac{- 5 - 3}{2 \cdot 4} : - 1

\frac{- 5 - 3}{2 \cdot 4}

数字相减:

- 5 - 3 = - 8

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

数字相乘:

2 \cdot 4 = 8

= \frac{- 8}{8}

使用分式法则:

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

= - \frac{8}{8}

使用法则

\frac{a}{a} = 1

= - 1

二次方程组的解是:

u = - \frac{1}{4}, u = - 1

u = - \frac{1}{4}, u = - 1

验证解

找到无定义的点（奇点）:

u = 0

取

5 + \frac{1}{u} + 4 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{1}{4}, u = - 1

u = csc (x)

代回

csc (x) = - \frac{1}{4}, csc (x) = - 1

csc (x) = - \frac{1}{4}, csc (x) = - 1

csc (x) = - \frac{1}{4} :

无解

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

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

无解

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

合并所有解

x = \frac{3 π}{2} + 2 πn

作图

流行的例子

cos(θ)= 3/5 (270360)

2sin(x)=tan(x)cos(x)-cos(x)

cos(θ)-sin(θ)=1,0<= θ<= 2pi