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受欢迎的三角函数 >

6sin(x)cos(x)+3sin(x)-2cos(x)-1=0

解答

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

解答

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

+1

度数

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 160.52877 \dots^{\circ} + 36 0^{\circ} n

求解步骤

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

分解

6 sin (x) cos (x) + 3 sin (x) - 2 cos (x) - 1 : (2 cos (x) + 1) (3 sin (x) - 1)

6 sin (x) cos (x) + 3 sin (x) - 2 cos (x) - 1

= (6 sin (x) cos (x) + 3 sin (x)) + (- 2 cos (x) - 1)

从

- 2 cos (x) - 1

分解出因式

- 1

:

- (2 cos (x) + 1)

- 2 cos (x) - 1

因式分解出通项

- 1

= - (2 cos (x) + 1)

从

6 sin (x) cos (x) + 3 sin (x)

分解出因式

3 sin (x)

:

3 sin (x) (2 cos (x) + 1)

6 sin (x) cos (x) + 3 sin (x)

将

6

改写为

2 \cdot 3

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

因式分解出通项

3 sin (x)

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

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

因式分解出通项

2 cos (x) + 1

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

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

分别求解每个部分

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

2 cos (x) + 1 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

2 cos (x) + 1 = 0

将

1

到右边

2 cos (x) + 1 = 0

两边减去

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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

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

3 sin (x) - 1 = 0

将

1

到右边

3 sin (x) - 1 = 0

两边加上

1

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

化简

3 sin (x) = 1

3 sin (x) = 1

两边除以

3

3 sin (x) = 1

两边除以

3

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

化简

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

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

使用反三角函数性质

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

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

的通解

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

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

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

合并所有解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arcsin (\frac{1}{3}) + 2 πn, x = π - arcsin (\frac{1}{3}) + 2 πn

以小数形式表示解

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 0.33983 \dots + 2 πn, x = π - 0.33983 \dots + 2 πn

作图

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