zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的三角函数 >

cot(x)+sqrt(2)cos(x)=0

解答

cot (x) + 2 cos (x) = 0

解答

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

+1

度数

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

cot (x) + 2 cos (x) = 0

用 sin, cos 表示

cot (x) + cos (x) 2

使用基本三角恒等式:

cot (x) = \frac{cos ( x )}{sin ( x )}

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

化简

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

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

将项转换为分式:

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

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

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

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

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

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

分解

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

cos (x) + cos (x) sin (x) 2

因式分解出通项

cos (x)

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

cos (x) (1 + 2 sin (x)) = 0

分别求解每个部分

cos (x) = 0 or 1 + 2 sin (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

1 + 2 sin (x) = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

1 + 2 sin (x) = 0

将

1

到右边

1 + 2 sin (x) = 0

两边减去

1

1 + 2 sin (x) - 1 = 0 - 1

化简

2 sin (x) = - 1

2 sin (x) = - 1

两边除以

2

2 sin (x) = - 1

两边除以

2

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

化简

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

化简

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

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

约分:

2

= sin (x)

化简

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

使用分式法则:

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

= - \frac{1}{2}

- \frac{1}{2}

有理化:

- \frac{2}{2}

- \frac{1}{2}

乘以共轭根式

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

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

的通解

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

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

合并所有解

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

作图

流行的例子

3cos(x)-sin(x)=1

3 cos (x) - sin (x) = 1

3 sin (θ) - 1 = 0

tan (x) = \frac{40}{9}

cos((3x)/2)= 1/2 ,-pi/2 <= x<= pi/2

cos (\frac{3 x}{2}) = \frac{1}{2}, - \frac{π}{2} \leq x \leq \frac{π}{2}

2cos(pi/2 t)=sqrt(3)

2 cos (\frac{π}{2} t) = 3