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受欢迎的三角函数 >

sin(2t)=2sin(t)

解答

sin (2 t) = 2 sin (t)

解答

t = 2 πn, t = π + 2 πn

+1

度数

t = 0^{\circ} + 36 0^{\circ} n, t = 18 0^{\circ} + 36 0^{\circ} n

求解步骤

sin (2 t) = 2 sin (t)

两边减去

2 sin (t)

sin (2 t) - 2 sin (t) = 0

使用三角恒等式改写

sin (2 t) - 2 sin (t)

使用倍角公式:

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

= 2 sin (t) cos (t) - 2 sin (t)

- 2 sin (t) + 2 cos (t) sin (t) = 0

分解

- 2 sin (t) + 2 cos (t) sin (t) : 2 sin (t) (cos (t) - 1)

- 2 sin (t) + 2 cos (t) sin (t)

因式分解出通项

2 sin (t)

= 2 sin (t) (- 1 + cos (t))

2 sin (t) (cos (t) - 1) = 0

分别求解每个部分

sin (t) = 0 or cos (t) - 1 = 0

sin (t) = 0 : t = 2 πn, t = π + 2 πn

sin (t) = 0

sin (t) = 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}

t = 0 + 2 πn, t = π + 2 πn

t = 0 + 2 πn, t = π + 2 πn

解

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn, t = π + 2 πn

cos (t) - 1 = 0 : t = 2 πn

cos (t) - 1 = 0

将

1

到右边

cos (t) - 1 = 0

两边加上

1

cos (t) - 1 + 1 = 0 + 1

化简

cos (t) = 1

cos (t) = 1

cos (t) = 1

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}

t = 0 + 2 πn

t = 0 + 2 πn

解

t = 0 + 2 πn : t = 2 πn

t = 0 + 2 πn

0 + 2 πn = 2 πn

t = 2 πn

t = 2 πn

合并所有解

t = 2 πn, t = π + 2 πn

作图

流行的例子

cos (α) = 0

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so l v e f or x, cos (x^{3} y^{3}) = 5 y^{3}

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cos^{2} (\frac{3}{4} π + x) + sin^{2} (\frac{7}{4} π - x) = 1

sin(4x)=-sin(x)

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