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2sin(x)+sqrt(3)tan(x)=0

解答

2 sin (x) + 3 tan (x) = 0

解答

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

求解步骤

2 sin (x) + 3 tan (x) = 0

用 sin, cos 表示

2 sin (x) + 3 tan (x)

使用基本三角恒等式:

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

= 2 sin (x) + 3 \frac{sin ( x )}{cos ( x )}

化简

2 sin (x) + 3 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x ) cos ( x ) + 3 sin ( x )}{cos ( x )}

2 sin (x) + 3 \frac{sin ( x )}{cos ( x )}

乘

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

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

分式相乘:

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

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

= 2 sin (x) + \frac{3 sin ( x )}{cos ( x )}

将项转换为分式:

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

= \frac{2 sin ( x ) cos ( x )}{cos ( x )} + \frac{sin ( x ) 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{2 sin ( x ) cos ( x ) + sin ( x ) 3}{cos ( x )}

= \frac{2 sin ( x ) cos ( x ) + 3 sin ( x )}{cos ( x )}

\frac{sin ( x ) 3 + 2 cos ( x ) sin ( x )}{cos ( x )} = 0

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

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

分解

sin (x) 3 + 2 cos (x) sin (x) : sin (x) (3 + 2 cos (x))

sin (x) 3 + 2 cos (x) sin (x)

因式分解出通项

sin (x)

= sin (x) (3 + 2 cos (x))

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

分别求解每个部分

sin (x) = 0 or 3 + 2 cos (x) = 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

3 + 2 cos (x) = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

3 + 2 cos (x) = 0

将

3

到右边

3 + 2 cos (x) = 0

两边减去

3

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

化简

2 cos (x) = - 3

2 cos (x) = - 3

两边除以

2

2 cos (x) = - 3

两边除以

2

\frac{2 cos ( x )}{2} = \frac{- 3}{2}

化简

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

合并所有解

x = 2 πn, x = π + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

流行的例子

tan(t)=2,0<t< pi/2 ,cos(t)

tan (t) = 2, 0 < t < \frac{π}{2}, cos (t)

(cos(x)+sin(x))/(sin(x))= 1/(tan(x))

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

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

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

sin(θ)= 1/4 ,0<θ< pi/2

sin (θ) = \frac{1}{4}, 0 < θ < \frac{π}{2}

sin(2θ)=(1.35*10)/((38)^2)

sin (2 θ) = \frac{1.35 \cdot 10}{( 38 ) ^{2}}