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受欢迎的三角函数 >

sqrt(3)sin(θ/2)+sin(θ)=0

解答

3 sin (\frac{θ}{2}) + sin (θ) = 0

解答

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

+1

度数

θ = 0^{\circ} + 72 0^{\circ} n, θ = 36 0^{\circ} + 72 0^{\circ} n, θ = 30 0^{\circ} + 72 0^{\circ} n, θ = 42 0^{\circ} + 72 0^{\circ} n

求解步骤

3 sin (\frac{θ}{2}) + sin (θ) = 0

令

: u = \frac{θ}{2}

3 sin (u) + sin (2 u) = 0

使用三角恒等式改写

sin (2 u) + sin (u) 3

使用倍角公式:

sin (2 x) = 2 sin (x) cos (x)

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

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

分解

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

sin (u) 3 + 2 cos (u) sin (u)

因式分解出通项

sin (u)

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

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

分别求解每个部分

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

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

解

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

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

3 + 2 cos (u) = 0

将

3

到右边

3 + 2 cos (u) = 0

两边减去

3

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

化简

2 cos (u) = - 3

2 cos (u) = - 3

两边除以

2

2 cos (u) = - 3

两边除以

2

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

化简

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

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

cos (u) = - \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}

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

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

合并所有解

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

u = \frac{θ}{2}

代回

\frac{θ}{2} = 2 πn : θ = 4 πn

\frac{θ}{2} = 2 πn

在两边乘以

2

\frac{θ}{2} = 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot 2 πn

化简

θ = 4 πn

θ = 4 πn

\frac{θ}{2} = π + 2 πn : θ = 2 π + 4 πn

\frac{θ}{2} = π + 2 πn

在两边乘以

2

\frac{θ}{2} = π + 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 π + 2 \cdot 2 πn

化简

θ = 2 π + 4 πn

θ = 2 π + 4 πn

\frac{θ}{2} = \frac{5 π}{6} + 2 πn : θ = \frac{5 π}{3} + 4 πn

\frac{θ}{2} = \frac{5 π}{6} + 2 πn

在两边乘以

2

\frac{θ}{2} = \frac{5 π}{6} + 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} = 2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

分式相乘:

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

= \frac{5 π 2}{6}

数字相乘:

5 \cdot 2 = 10

= \frac{10 π}{6}

约分:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

θ = \frac{5 π}{3} + 4 πn

θ = \frac{5 π}{3} + 4 πn

θ = \frac{5 π}{3} + 4 πn

\frac{θ}{2} = \frac{7 π}{6} + 2 πn : θ = \frac{7 π}{3} + 4 πn

\frac{θ}{2} = \frac{7 π}{6} + 2 πn

在两边乘以

2

\frac{θ}{2} = \frac{7 π}{6} + 2 πn

在两边乘以

2

\frac{2 θ}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} = 2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

化简

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

数字相除:

\frac{2}{2} = 1

= θ

化简

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

分式相乘:

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

= \frac{7 π 2}{6}

数字相乘:

7 \cdot 2 = 14

= \frac{14 π}{6}

约分:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

数字相乘:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

θ = \frac{7 π}{3} + 4 πn

θ = \frac{7 π}{3} + 4 πn

θ = \frac{7 π}{3} + 4 πn

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

作图

流行的例子

2sin(x)cos(x)sin(x)=cos(x)

2 sin (x) cos (x) sin (x) = cos (x)

-6csc^2(x)+10cot(x)=-10

- 6 csc^{2} (x) + 10 cot (x) = - 10

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

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

sin(x)= 1/2 ,cos(x)=-(sqrt(3))/2

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

cos(x)=(2sqrt(5))/5

cos (x) = \frac{2 5}{5}