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受欢迎的三角函数 >

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

解答

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

解答

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

+1

度数

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

求解步骤

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

两边减去

6 cos (x)

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

使用三角恒等式改写

- 6 cos (x) + 6 sin (2 x)

使用倍角公式:

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

= - 6 cos (x) + 6 \cdot 2 sin (x) cos (x)

化简

= - 6 cos (x) + 12 sin (x) cos (x)

- 6 cos (x) + 12 cos (x) sin (x) = 0

分解

- 6 cos (x) + 12 cos (x) sin (x) : 6 cos (x) (2 sin (x) - 1)

- 6 cos (x) + 12 cos (x) sin (x)

将

12

改写为

2 \cdot 6

= - 6 cos (x) + 2 \cdot 6 sin (x) cos (x)

因式分解出通项

6 cos (x)

= 6 cos (x) (- 1 + 2 sin (x))

6 cos (x) (2 sin (x) - 1) = 0

分别求解每个部分

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

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

cos (x) = 0

cos (x) = 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}

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

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

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

2 sin (x) - 1 = 0

将

1

到右边

2 sin (x) - 1 = 0

两边加上

1

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

化简

2 sin (x) = 1

2 sin (x) = 1

两边除以

2

2 sin (x) = 1

两边除以

2

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

化简

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

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

sin (x) = \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}

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

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

合并所有解

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

作图

流行的例子

cot (θ) = \frac{5}{12}

tan(x)=(7.4)/(9.3)

tan (x) = \frac{7.4}{9.3}

tan^{4} (x) - 1 = 0

15sin(x)+19=14sin(x)+18

15 sin (x) + 19 = 14 sin (x) + 18

3+3sin(θ)= 6/(3-2sin(θ))

3 + 3 sin (θ) = \frac{6}{3 - 2 sin ( θ )}