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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

- sin (2 x) + sin (x) 2

使用倍角公式:

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

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

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

分解

sin (x) 2 - 2 cos (x) sin (x) : sin (x) (2 - 2 cos (x))

sin (x) 2 - 2 cos (x) sin (x)

因式分解出通项

sin (x)

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

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

分别求解每个部分

sin (x) = 0 or 2 - 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

2 - 2 cos (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

2 - 2 cos (x) = 0

将

2

到右边

2 - 2 cos (x) = 0

两边减去

2

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

化简

- 2 cos (x) = - 2

- 2 cos (x) = - 2

两边除以

- 2

- 2 cos (x) = - 2

两边除以

- 2

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

化简

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

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

cos (x) = \frac{2}{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{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

合并所有解

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

作图

流行的例子

sin(2θ)=-sqrt(3)cos(2θ)

sin(2x)+6cos(x)=0

cos(3x)= 1/2 ,0<= x<= 2pi

2cos^2(x)-cos(x)=0,0<= x<2pi