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受欢迎的三角函数 >

2sin(x)=csc(x)-6,0<= x<2pi

解答

2 sin (x) = csc (x) - 6, 0 \leq x < 2 π

解答

x = 0.15898 \dots, x = π - 0.15898 \dots

+1

度数

x = 9.10895 \dots^{\circ}, x = 170.89104 \dots^{\circ}

求解步骤

2 sin (x) = csc (x) - 6, 0 \leq x < 2 π

两边减去

csc (x) - 6

2 sin (x) - csc (x) + 6 = 0

使用三角恒等式改写

6 - csc (x) + 2 sin (x)

使用基本三角恒等式:

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

= 6 - csc (x) + 2 \cdot \frac{1}{csc ( x )}

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

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

分式相乘:

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

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

数字相乘:

1 \cdot 2 = 2

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

= 6 - csc (x) + \frac{2}{csc ( x )}

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

用替代法求解

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

令

: csc (x) = u

6 - u + \frac{2}{u} = 0

6 - u + \frac{2}{u} = 0 : u = 3 - 11, u = 3 + 11

6 - u + \frac{2}{u} = 0

在两边乘以

u

6 - u + \frac{2}{u} = 0

在两边乘以

u

6 u - uu + \frac{2}{u} u = 0 \cdot u

化简

6 u - uu + \frac{2}{u} u = 0 \cdot u

化简

- uu : - u^{2}

- uu

使用指数法则:

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

uu = u^{1 + 1}

= - u^{1 + 1}

数字相加:

1 + 1 = 2

= - u^{2}

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

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

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

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

解

6 u - u^{2} + 2 = 0 : u = 3 - 11, u = 3 + 11

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

改写成标准形式

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

- u^{2} + 6 u + 2 = 0

使用求根公式求解

- u^{2} + 6 u + 2 = 0

二次方程求根公式:

若

a = - 1, b = 6, c = 2

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

6^{2} - 4 (- 1) \cdot 2 = 211

6^{2} - 4 (- 1) \cdot 2

使用法则

- (- a) = a

= 6^{2} + 4 \cdot 1 \cdot 2

数字相乘:

4 \cdot 1 \cdot 2 = 8

= 6^{2} + 8

6^{2} = 36

= 36 + 8

数字相加:

36 + 8 = 44

= 44

44

质因数分解:

2^{2} \cdot 11

44

44

除以

244 = 22 \cdot 2

= 2 \cdot 22

22

除以

222 = 11 \cdot 2

= 2 \cdot 2 \cdot 11

2, 11

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 11

= 2^{2} \cdot 11

= 2^{2} \cdot 11

使用根式运算法则:

n ab = n a n b

= 11 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 211

u_{1, 2} = \frac{- 6 \pm 2 11}{2 ( - 1 )}

将解分隔开

u_{1} = \frac{- 6 + 2 11}{2 ( - 1 )}, u_{2} = \frac{- 6 - 2 11}{2 ( - 1 )}

u = \frac{- 6 + 2 11}{2 ( - 1 )} : 3 - 11

\frac{- 6 + 2 11}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 6 + 2 11}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 + 2 11}{- 2}

使用分式法则:

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

= - \frac{- 6 + 2 11}{2}

消掉

\frac{- 6 + 2 11}{2} : 11 - 3

\frac{- 6 + 2 11}{2}

分解

- 6 + 211 : 2 (- 3 + 11)

- 6 + 211

改写为

= - 2 \cdot 3 + 211

因式分解出通项

2

= 2 (- 3 + 11)

= \frac{2 ( - 3 + 11 )}{2}

数字相除:

\frac{2}{2} = 1

= - 3 + 11

= - (11 - 3)

打开括号

= - (- 3) - (11)

使用加减运算法则

- (- a) = a, - (a) = - a

= 3 - 11

u = \frac{- 6 - 2 11}{2 ( - 1 )} : 3 + 11

\frac{- 6 - 2 11}{2 ( - 1 )}

去除括号:

(- a) = - a

= \frac{- 6 - 2 11}{- 2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 6 - 2 11}{- 2}

使用分式法则:

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

- 6 - 211 = - (6 + 211)

= \frac{6 + 2 11}{2}

分解

6 + 211 : 2 (3 + 11)

6 + 211

改写为

= 2 \cdot 3 + 211

因式分解出通项

2

= 2 (3 + 11)

= \frac{2 ( 3 + 11 )}{2}

数字相除:

\frac{2}{2} = 1

= 3 + 11

二次方程组的解是:

u = 3 - 11, u = 3 + 11

u = 3 - 11, u = 3 + 11

验证解

找到无定义的点（奇点）:

u = 0

取

6 - u + \frac{2}{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 = 3 - 11, u = 3 + 11

u = csc (x)

代回

csc (x) = 3 - 11, csc (x) = 3 + 11

csc (x) = 3 - 11, csc (x) = 3 + 11

csc (x) = 3 - 11, 0 \leq x < 2 π :

无解

csc (x) = 3 - 11, 0 \leq x < 2 π

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

无解

csc (x) = 3 + 11, 0 \leq x < 2 π : x = arccsc (3 + 11), x = π - arccsc (3 + 11)

csc (x) = 3 + 11, 0 \leq x < 2 π

使用反三角函数性质

csc (x) = 3 + 11

csc (x) = 3 + 11

的通解

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

x = arccsc (3 + 11) + 2 πn, x = π - arccsc (3 + 11) + 2 πn

x = arccsc (3 + 11) + 2 πn, x = π - arccsc (3 + 11) + 2 πn

在

0 \leq x < 2 π

范围内的解

x = arccsc (3 + 11), x = π - arccsc (3 + 11)

合并所有解

x = arccsc (3 + 11), x = π - arccsc (3 + 11)

以小数形式表示解

x = 0.15898 \dots, x = π - 0.15898 \dots

作图

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