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受欢迎的 >

sin(x)+sin(2x)=0,0<= x<= 2pi

解答

sin (x) + sin (2 x) = 0, 0 \leq x \leq 2 π

解答

x = 0, x = π, x = 2 π, x = \frac{2 π}{3}, x = \frac{4 π}{3}

+1

度数

x = 0^{\circ}, x = 18 0^{\circ}, x = 36 0^{\circ}, x = 12 0^{\circ}, x = 24 0^{\circ}

求解步骤

sin (x) + sin (2 x) = 0, 0 \leq x \leq 2 π

使用三角恒等式改写

sin (2 x) + sin (x)

使用倍角公式:

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

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

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

分解

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

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

因式分解出通项

sin (x)

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

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

分别求解每个部分

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

sin (x) = 0, 0 \leq x \leq 2 π : x = 0, x = π, x = 2 π

sin (x) = 0, 0 \leq x \leq 2 π

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

在

0 \leq x \leq 2 π

范围内的解

x = 0, x = π, x = 2 π

2 cos (x) + 1 = 0, 0 \leq x \leq 2 π : x = \frac{2 π}{3}, x = \frac{4 π}{3}

2 cos (x) + 1 = 0, 0 \leq x \leq 2 π

将

1

到右边

2 cos (x) + 1 = 0

两边减去

1

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

化简

2 cos (x) = - 1

2 cos (x) = - 1

两边除以

2

2 cos (x) = - 1

两边除以

2

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

化简

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

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

在

0 \leq x \leq 2 π

范围内的解

x = \frac{2 π}{3}, x = \frac{4 π}{3}

合并所有解

x = 0, x = π, x = 2 π, x = \frac{2 π}{3}, x = \frac{4 π}{3}

流行的例子

8sin(2x)=16cos(x)

8 sin (2 x) = 16 cos (x)

3 arccos (x) = π

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

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

2 cos (4 x) + 5 = 6

cos(2θ)=-(sqrt(3))/2 ,0<= θ<= 2pi

cos (2 θ) = - \frac{3}{2}, 0 \leq θ \leq 2 π