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 >

3tan(x)=sin(x)

解答

3 tan (x) = sin (x)

解答

x = 2 πn, x = π + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

3 tan (x) = sin (x)

两边减去

sin (x)

3 tan (x) - sin (x) = 0

用 sin, cos 表示

- sin (x) + 3 tan (x)

使用基本三角恒等式:

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

= - sin (x) + 3 \cdot \frac{sin ( x )}{cos ( x )}

化简

- sin (x) + 3 \cdot \frac{sin ( x )}{cos ( x )} : \frac{- sin ( x ) cos ( x ) + 3 sin ( x )}{cos ( x )}

- sin (x) + 3 \cdot \frac{sin ( x )}{cos ( x )}

乘

3 \cdot \frac{sin ( x )}{cos ( x )} : \frac{3 sin ( x )}{cos ( x )}

3 \cdot \frac{sin ( x )}{cos ( x )}

分式相乘:

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

= \frac{sin ( x ) \cdot 3}{cos ( x )}

= - sin (x) + \frac{3 sin ( x )}{cos ( x )}

将项转换为分式:

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

= - \frac{sin ( x ) cos ( x )}{cos ( x )} + \frac{sin ( x ) \cdot 3}{cos ( 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{- sin ( x ) cos ( x ) + sin ( x ) \cdot 3}{cos ( x )}

= \frac{- sin ( x ) cos ( x ) + 3 sin ( x )}{cos ( x )}

\frac{3 sin ( x ) - cos ( x ) sin ( x )}{cos ( x )} = 0

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

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

分解

3 sin (x) - cos (x) sin (x) : - sin (x) (cos (x) - 3)

3 sin (x) - cos (x) sin (x)

因式分解出通项

- sin (x)

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

- sin (x) (cos (x) - 3) = 0

分别求解每个部分

sin (x) = 0 or cos (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

cos (x) - 3 = 0 :

无解

cos (x) - 3 = 0

将

3

到右边

cos (x) - 3 = 0

两边加上

3

cos (x) - 3 + 3 = 0 + 3

化简

cos (x) = 3

cos (x) = 3

- 1 \leq cos (x) \leq 1

无解

合并所有解

x = 2 πn, x = π + 2 πn

作图

流行的例子

5 = sinh (x)

- 16 cos (2 x) = 0

6sqrt(2)+12cos(x-45)=0

62 + 12 cos (x - 4 5^{\circ}) = 0

(2*9.8*l(1-cos(θ)))=0.018

(2 \cdot 9.8 \cdot l (1 - cos (θ))) = 0.018

\frac{1}{tan ( x )} = 0.5