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受欢迎的三角函数 >

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

解答

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

解答

x = 0, x = π, x = - π, x = \frac{π}{3}, x = - \frac{π}{3}

+1

度数

x = 0^{\circ}, x = 18 0^{\circ}, x = - 18 0^{\circ}, x = 6 0^{\circ}, x = - 6 0^{\circ}

求解步骤

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

sin (x) = 0, - π \leq x \leq 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

- π \leq x \leq π

范围内的解

x = 0, x = π, x = - π

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

2 cos (x) - 1 = 0, - π \leq x \leq π

将

1

到右边

2 cos (x) - 1 = 0

两边加上

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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

在

- π \leq x \leq π

范围内的解

x = \frac{π}{3}, x = - \frac{π}{3}

合并所有解

x = 0, x = π, x = - π, x = \frac{π}{3}, x = - \frac{π}{3}

作图

流行的例子

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2 (tan (x) + 3) = 5 + tan (x)

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3cos^2(x)-cos(2x)=1

3 cos^{2} (x) - cos (2 x) = 1

sin(2x)+sin(4x)=cos(x)

sin (2 x) + sin (4 x) = cos (x)

2 sin (x) = - 0.684