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受欢迎的三角函数 >

16sin(3x)cos(x)+8sqrt(3)sin(3x)-2cos(x)-sqrt(3)=0

解答

16 sin (3 x) cos (x) + 83 sin (3 x) - 2 cos (x) - 3 = 0

解答

x = \frac{0.12532 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{0.12532 \dots}{3} + \frac{2 πn}{3}, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

+1

度数

x = 2.39358 \dots^{\circ} + 12 0^{\circ} n, x = 57.60641 \dots^{\circ} + 12 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

求解步骤

16 sin (3 x) cos (x) + 83 sin (3 x) - 2 cos (x) - 3 = 0

分解

16 sin (3 x) cos (x) + 83 sin (3 x) - 2 cos (x) - 3 : (- 1 + 8 sin (3 x)) (2 cos (x) + 3)

16 sin (3 x) cos (x) + 83 sin (3 x) - 2 cos (x) - 3

分解

16 sin (3 x) cos (x) - 2 cos (x) : 2 cos (x) (8 sin (3 x) - 1)

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

改写为

= 8 \cdot 2 cos (x) sin (3 x) - 1 \cdot 2 cos (x)

因式分解出通项

2 cos (x)

= 2 cos (x) (8 sin (3 x) - 1)

分解

83 sin (3 x) - 3 : 3 (8 sin (3 x) - 1)

83 sin (3 x) - 3

因式分解出通项

3

= 3 (8 sin (3 x) - 1)

= 2 cos (x) (8 sin (3 x) - 1) + 3 (8 sin (3 x) - 1)

因式分解出通项

(- 1 + 8 sin (3 x))

= (- 1 + 8 sin (3 x)) (2 cos (x) + 3)

(- 1 + 8 sin (3 x)) (2 cos (x) + 3) = 0

分别求解每个部分

- 1 + 8 sin (3 x) = 0 or 2 cos (x) + 3 = 0

- 1 + 8 sin (3 x) = 0 : x = \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

- 1 + 8 sin (3 x) = 0

将

1

到右边

- 1 + 8 sin (3 x) = 0

两边加上

1

- 1 + 8 sin (3 x) + 1 = 0 + 1

化简

8 sin (3 x) = 1

8 sin (3 x) = 1

两边除以

8

8 sin (3 x) = 1

两边除以

8

\frac{8 sin ( 3 x )}{8} = \frac{1}{8}

化简

sin (3 x) = \frac{1}{8}

sin (3 x) = \frac{1}{8}

使用反三角函数性质

sin (3 x) = \frac{1}{8}

sin (3 x) = \frac{1}{8}

的通解

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

3 x = arcsin (\frac{1}{8}) + 2 πn, 3 x = π - arcsin (\frac{1}{8}) + 2 πn

3 x = arcsin (\frac{1}{8}) + 2 πn, 3 x = π - arcsin (\frac{1}{8}) + 2 πn

解

3 x = arcsin (\frac{1}{8}) + 2 πn : x = \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

3 x = arcsin (\frac{1}{8}) + 2 πn

两边除以

3

3 x = arcsin (\frac{1}{8}) + 2 πn

两边除以

3

\frac{3 x}{3} = \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

化简

x = \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

x = \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

解

3 x = π - arcsin (\frac{1}{8}) + 2 πn : x = \frac{π}{3} - \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

3 x = π - arcsin (\frac{1}{8}) + 2 πn

两边除以

3

3 x = π - arcsin (\frac{1}{8}) + 2 πn

两边除以

3

\frac{3 x}{3} = \frac{π}{3} - \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

化简

x = \frac{π}{3} - \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

x = \frac{π}{3} - \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

x = \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}

2 cos (x) + 3 = 0 : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

2 cos (x) + 3 = 0

将

3

到右边

2 cos (x) + 3 = 0

两边减去

3

2 cos (x) + 3 - 3 = 0 - 3

化简

2 cos (x) = - 3

2 cos (x) = - 3

两边除以

2

2 cos (x) = - 3

两边除以

2

\frac{2 cos ( x )}{2} = \frac{- 3}{2}

化简

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

合并所有解

x = \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{arcsin ( \frac{1}{8} )}{3} + \frac{2 πn}{3}, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

以小数形式表示解

x = \frac{0.12532 \dots}{3} + \frac{2 πn}{3}, x = \frac{π}{3} - \frac{0.12532 \dots}{3} + \frac{2 πn}{3}, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

作图

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