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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

求解步骤

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

用 sin, cos 表示

2 sin (x) + \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} = 0

化简

2 sin (x) + \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} : \frac{2 sin ^{2} ( x ) + cos ( x ) - 1}{sin ( x )}

2 sin (x) + \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )}

合并分式

\frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} : \frac{cos ( x ) - 1}{sin ( x )}

使用法则

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

= \frac{cos ( x ) - 1}{sin ( x )}

= 2 sin (x) + \frac{cos ( x ) - 1}{sin ( x )}

将项转换为分式:

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

= \frac{2 sin ( x ) sin ( x )}{sin ( x )} + \frac{cos ( x ) - 1}{sin ( x )}

因为分母相等，所以合并分式:

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

= \frac{2 sin ( x ) sin ( x ) + cos ( x ) - 1}{sin ( x )}

2 sin (x) sin (x) + cos (x) - 1 = 2 sin^{2} (x) + cos (x) - 1

2 sin (x) sin (x) + cos (x) - 1

2 sin (x) sin (x) = 2 sin^{2} (x)

2 sin (x) sin (x)

使用指数法则:

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

sin (x) sin (x) = sin^{1 + 1} (x)

= 2 sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= 2 sin^{2} (x)

= 2 sin^{2} (x) + cos (x) - 1

= \frac{2 sin ^{2} ( x ) + cos ( x ) - 1}{sin ( x )}

\frac{2 sin ^{2} ( x ) + cos ( x ) - 1}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin^{2} (x) + cos (x) - 1 = 0

两边减去

cos (x)

2 sin^{2} (x) - 1 = - cos (x)

两边进行平方

(2 sin^{2} (x) - 1)^{2} = (- cos (x))^{2}

两边减去

(- cos (x))^{2}

(2 sin^{2} (x) - 1)^{2} - cos^{2} (x) = 0

分解

(2 sin^{2} (x) - 1)^{2} - cos^{2} (x) : (2 sin^{2} (x) - 1 + cos (x)) (2 sin^{2} (x) - 1 - cos (x))

(2 sin^{2} (x) - 1)^{2} - cos^{2} (x)

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(2 sin^{2} (x) - 1)^{2} - cos^{2} (x) = ((2 sin^{2} (x) - 1) + cos (x)) ((2 sin^{2} (x) - 1) - cos (x))

= ((2 sin^{2} (x) - 1) + cos (x)) ((2 sin^{2} (x) - 1) - cos (x))

整理后得

= (2 sin^{2} (x) + cos (x) - 1) (2 sin^{2} (x) - cos (x) - 1)

(2 sin^{2} (x) - 1 + cos (x)) (2 sin^{2} (x) - 1 - cos (x)) = 0

分别求解每个部分

2 sin^{2} (x) - 1 + cos (x) = 0 or 2 sin^{2} (x) - 1 - cos (x) = 0

2 sin^{2} (x) - 1 + cos (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 2 πn

2 sin^{2} (x) - 1 + cos (x) = 0

使用三角恒等式改写

- 1 + cos (x) + 2 sin^{2} (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 1 + cos (x) + 2 (1 - cos^{2} (x))

化简

- 1 + cos (x) + 2 (1 - cos^{2} (x)) : cos (x) - 2 cos^{2} (x) + 1

- 1 + cos (x) + 2 (1 - cos^{2} (x))

乘开

2 (1 - cos^{2} (x)) : 2 - 2 cos^{2} (x)

2 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (x)

= - 1 + cos (x) + 2 - 2 cos^{2} (x)

化简

- 1 + cos (x) + 2 - 2 cos^{2} (x) : cos (x) - 2 cos^{2} (x) + 1

- 1 + cos (x) + 2 - 2 cos^{2} (x)

对同类项分组

= cos (x) - 2 cos^{2} (x) - 1 + 2

数字相加/相减:

- 1 + 2 = 1

= cos (x) - 2 cos^{2} (x) + 1

= cos (x) - 2 cos^{2} (x) + 1

= cos (x) - 2 cos^{2} (x) + 1

1 + cos (x) - 2 cos^{2} (x) = 0

用替代法求解

1 + cos (x) - 2 cos^{2} (x) = 0

令

: cos (x) = u

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

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

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

改写成标准形式

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

- 2 u^{2} + u + 1 = 0

使用求根公式求解

- 2 u^{2} + u + 1 = 0

二次方程求根公式:

若

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

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

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

1^{2} - 4 (- 2) \cdot 1 = 3

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

使用法则

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 2) \cdot 1

使用法则

- (- a) = a

= 1 + 4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 1 + 8

数字相加:

1 + 8 = 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{- 1 \pm 3}{2 ( - 2 )}

将解分隔开

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

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

\frac{- 1 + 3}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 1 + 3}{- 2 \cdot 2}

数字相加/相减:

- 1 + 3 = 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{2}{- 4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

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

\frac{- 1 - 3}{2 ( - 2 )}

去除括号:

(- a) = - a

= \frac{- 1 - 3}{- 2 \cdot 2}

数字相减:

- 1 - 3 = - 4

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 4}{- 4}

使用分式法则:

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

= \frac{4}{4}

使用法则

\frac{a}{a} = 1

= 1

二次方程组的解是:

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

u = cos (x)

代回

cos (x) = - \frac{1}{2}, cos (x) = 1

cos (x) = - \frac{1}{2}, cos (x) = 1

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

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

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

cos (x) = 1 : x = 2 πn

cos (x) = 1

cos (x) = 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

解

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

合并所有解

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

2 sin^{2} (x) - 1 - cos (x) = 0 : x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 sin^{2} (x) - 1 - cos (x) = 0

使用三角恒等式改写

- 1 - cos (x) + 2 sin^{2} (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= - 1 - cos (x) + 2 (1 - cos^{2} (x))

化简

- 1 - cos (x) + 2 (1 - cos^{2} (x)) : - 2 cos^{2} (x) - cos (x) + 1

- 1 - cos (x) + 2 (1 - cos^{2} (x))

乘开

2 (1 - cos^{2} (x)) : 2 - 2 cos^{2} (x)

2 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

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

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

数字相乘:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (x)

= - 1 - cos (x) + 2 - 2 cos^{2} (x)

化简

- 1 - cos (x) + 2 - 2 cos^{2} (x) : - 2 cos^{2} (x) - cos (x) + 1

- 1 - cos (x) + 2 - 2 cos^{2} (x)

对同类项分组

= - cos (x) - 2 cos^{2} (x) - 1 + 2

数字相加/相减:

- 1 + 2 = 1

= - 2 cos^{2} (x) - cos (x) + 1

= - 2 cos^{2} (x) - cos (x) + 1

= - 2 cos^{2} (x) - cos (x) + 1

1 - cos (x) - 2 cos^{2} (x) = 0

用替代法求解

1 - cos (x) - 2 cos^{2} (x) = 0

令

: cos (x) = u

1 - u - 2 u^{2} = 0

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

1 - u - 2 u^{2} = 0

改写成标准形式

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

- 2 u^{2} - u + 1 = 0

使用求根公式求解

- 2 u^{2} - u + 1 = 0

二次方程求根公式:

若

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

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

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

(- 1)^{2} - 4 (- 2) \cdot 1 = 3

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

使用法则

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

使用指数法则:

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

若

n

是偶数

(- 1)^{2} = 1^{2}

= 1^{2}

使用法则

1^{a} = 1

= 1

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

数字相乘:

4 \cdot 2 \cdot 1 = 8

= 8

= 1 + 8

数字相加:

1 + 8 = 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{- ( - 1 ) \pm 3}{2 ( - 2 )}

将解分隔开

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

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

\frac{- ( - 1 ) + 3}{2 ( - 2 )}

去除括号:

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

= \frac{1 + 3}{- 2 \cdot 2}

数字相加:

1 + 3 = 4

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

数字相乘:

2 \cdot 2 = 4

= \frac{4}{- 4}

使用分式法则:

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

= - \frac{4}{4}

使用法则

\frac{a}{a} = 1

= - 1

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

\frac{- ( - 1 ) - 3}{2 ( - 2 )}

去除括号:

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

= \frac{1 - 3}{- 2 \cdot 2}

数字相减:

1 - 3 = - 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{- 4}

使用分式法则:

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

= \frac{2}{4}

约分:

2

= \frac{1}{2}

二次方程组的解是:

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

u = cos (x)

代回

cos (x) = - 1, cos (x) = \frac{1}{2}

cos (x) = - 1, cos (x) = \frac{1}{2}

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

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

的通解

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

合并所有解

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

合并所有解

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

将解代入原方程进行验证

将它们代入

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

检验解是否符合

去除与方程不符的解。

检验

\frac{2 π}{3} + 2 πn

的解:

真

\frac{2 π}{3} + 2 πn

代入

n = 1

\frac{2 π}{3} + 2 π 1

对于

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

代入

x = \frac{2 π}{3} + 2 π 1

2 sin (\frac{2 π}{3} + 2 π 1) + cot (\frac{2 π}{3} + 2 π 1) - csc (\frac{2 π}{3} + 2 π 1) = 0

整理后得

0 = 0

\Rightarrow 真

检验

\frac{4 π}{3} + 2 πn

的解:

真

\frac{4 π}{3} + 2 πn

代入

n = 1

\frac{4 π}{3} + 2 π 1

对于

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

代入

x = \frac{4 π}{3} + 2 π 1

2 sin (\frac{4 π}{3} + 2 π 1) + cot (\frac{4 π}{3} + 2 π 1) - csc (\frac{4 π}{3} + 2 π 1) = 0

整理后得

0 = 0

\Rightarrow 真

检验

2 πn

的解:

假

2 πn

代入

n = 1

2 π 1

对于

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

代入

x = 2 π 1

2 sin (2 π 1) + cot (2 π 1) - csc (2 π 1) = 0

未定义

\Rightarrow 假

检验

π + 2 πn

的解:

假

π + 2 πn

代入

n = 1

π + 2 π 1

对于

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

代入

x = π + 2 π 1

2 sin (π + 2 π 1) + cot (π + 2 π 1) - csc (π + 2 π 1) = 0

整理后得

- \infty = 0

\Rightarrow 假

检验

\frac{π}{3} + 2 πn

的解:

假

\frac{π}{3} + 2 πn

代入

n = 1

\frac{π}{3} + 2 π 1

对于

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

代入

x = \frac{π}{3} + 2 π 1

2 sin (\frac{π}{3} + 2 π 1) + cot (\frac{π}{3} + 2 π 1) - csc (\frac{π}{3} + 2 π 1) = 0

整理后得

1.15470 \dots = 0

\Rightarrow 假

检验

\frac{5 π}{3} + 2 πn

的解:

假

\frac{5 π}{3} + 2 πn

代入

n = 1

\frac{5 π}{3} + 2 π 1

对于

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

代入

x = \frac{5 π}{3} + 2 π 1

2 sin (\frac{5 π}{3} + 2 π 1) + cot (\frac{5 π}{3} + 2 π 1) - csc (\frac{5 π}{3} + 2 π 1) = 0

整理后得

- 1.15470 \dots = 0

\Rightarrow 假

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

作图

流行的例子

cosh (x) = \frac{5}{3}

2cos^2(x)-3sin(x)+2=0

2 cos^{2} (x) - 3 sin (x) + 2 = 0

sin (θ) = 0.75

5 cos (x) - 5 = 0

-2sin(x)cos(x)=0

- 2 sin (x) cos (x) = 0