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受欢迎的三角函数 >

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

解答

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

解答

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

+1

度数

t = 9 0^{\circ} + 36 0^{\circ} n, t = 27 0^{\circ} + 36 0^{\circ} n, t = 3 0^{\circ} + 36 0^{\circ} n, t = 15 0^{\circ} + 36 0^{\circ} n

求解步骤

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

使用三角恒等式改写

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

使用倍角公式:

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

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

化简

= 2 cos (t) - 4 sin (t) cos (t)

2 cos (t) - 4 cos (t) sin (t) = 0

分解

2 cos (t) - 4 cos (t) sin (t) : - 2 cos (t) (2 sin (t) - 1)

2 cos (t) - 4 cos (t) sin (t)

将

- 4

改写为

2 \cdot 2

= 2 cos (t) + 2 \cdot 2 sin (t) cos (t)

因式分解出通项

- 2 cos (t)

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

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

分别求解每个部分

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

cos (t) = 0 : t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

cos (t) = 0

cos (t) = 0

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 = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn

2 sin (t) - 1 = 0 : t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

2 sin (t) - 1 = 0

将

1

到右边

2 sin (t) - 1 = 0

两边加上

1

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

化简

2 sin (t) = 1

2 sin (t) = 1

两边除以

2

2 sin (t) = 1

两边除以

2

\frac{2 sin ( t )}{2} = \frac{1}{2}

化简

sin (t) = \frac{1}{2}

sin (t) = \frac{1}{2}

sin (t) = \frac{1}{2}

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

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

合并所有解

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

作图

流行的例子

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

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

3 sin (2 x) = 1.5

sin(2x-50)=-1/2

sin (2 x - 5 0^{\circ}) = - \frac{1}{2}

cos(x)=cos(3*x)+2*sin(2*x)

cos (x) = cos (3 \cdot x) + 2 \cdot sin (2 \cdot x)

sqrt(3)sin(x)-cos(x)=sqrt(2)

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