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受欢迎的三角函数 >

sin(3x)cos(3x)-1/4 >0

解答

sin (3 x) cos (3 x) - \frac{1}{4} > 0

解答

\frac{π}{36} + \frac{π}{3} n < x < \frac{5 π}{36} + \frac{π}{3} n

+2

间隔符号

(\frac{π}{36} + \frac{π}{3} n, \frac{5 π}{36} + \frac{π}{3} n)

十进制

0.08726 \dots + \frac{π}{3} n < x < 0.43633 \dots + \frac{π}{3} n

求解步骤

sin (3 x) cos (3 x) - \frac{1}{4} > 0

利用以下特性:

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

因此

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

- \frac{1}{4} + \frac{sin ( 2 \cdot 3 x )}{2} > 0

化简

- \frac{1}{4} + \frac{sin ( 2 \cdot 3 x )}{2} : - \frac{1}{4} + \frac{1}{2} sin (6 x)

- \frac{1}{4} + \frac{sin ( 2 \cdot 3 x )}{2}

数字相乘:

2 \cdot 3 = 6

= - \frac{1}{4} + \frac{sin ( 6 x )}{2}

= - \frac{1}{4} + \frac{1}{2} sin (6 x)

- \frac{1}{4} + \frac{1}{2} sin (6 x) > 0

将

\frac{1}{4}

到右边

- \frac{1}{4} + \frac{1}{2} sin (6 x) > 0

两边加上

\frac{1}{4}

- \frac{1}{4} + \frac{1}{2} sin (6 x) + \frac{1}{4} > 0 + \frac{1}{4}

化简

\frac{1}{2} sin (6 x) > \frac{1}{4}

\frac{1}{2} sin (6 x) > \frac{1}{4}

在两边乘以

2

\frac{1}{2} sin (6 x) > \frac{1}{4}

在两边乘以

2

2 \cdot \frac{1}{2} sin (6 x) > \frac{1 \cdot 2}{4}

化简

2 \cdot \frac{1}{2} sin (6 x) > \frac{1 \cdot 2}{4}

化简

2 \cdot \frac{1}{2} sin (6 x) : sin (6 x)

2 \cdot \frac{1}{2} sin (6 x)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2} sin (6 x)

约分:

2

= sin (6 x) \cdot 1

乘以:

sin (6 x) \cdot 1 = sin (6 x)

= sin (6 x)

化简

\frac{1 \cdot 2}{4} : \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

sin (6 x) > \frac{1}{2}

sin (6 x) > \frac{1}{2}

sin (6 x) > \frac{1}{2}

对于

sin (x) > a

，若

- 1 \leq a < 1

，则

arcsin (a) + 2 πn < x < π - arcsin (a) + 2 πn

arcsin (\frac{1}{2}) + 2 πn < 6 x < π - arcsin (\frac{1}{2}) + 2 πn

若

a < u < b

，则

a < u and u < b

arcsin (\frac{1}{2}) + 2 πn < 6 x and 6 x < π - arcsin (\frac{1}{2}) + 2 πn

arcsin (\frac{1}{2}) + 2 πn < 6 x : x > \frac{π}{36} + \frac{πn}{3}

arcsin (\frac{1}{2}) + 2 πn < 6 x

交换两边

6 x > arcsin (\frac{1}{2}) + 2 πn

化简

arcsin (\frac{1}{2}) + 2 πn : \frac{π}{6} + 2 πn

arcsin (\frac{1}{2}) + 2 πn

使用以下普通恒等式:

arcsin (\frac{1}{2}) = \frac{π}{6}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{6} + 2 πn

6 x > \frac{π}{6} + 2 πn

两边除以

6

6 x > \frac{π}{6} + 2 πn

两边除以

6

\frac{6 x}{6} > \frac{\frac{π}{6}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} > \frac{\frac{π}{6}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} : x

\frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{\frac{π}{6}}{6} + \frac{2 πn}{6} : \frac{π}{36} + \frac{πn}{3}

\frac{\frac{π}{6}}{6} + \frac{2 πn}{6}

\frac{\frac{π}{6}}{6} = \frac{π}{36}

\frac{\frac{π}{6}}{6}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 6}

数字相乘:

6 \cdot 6 = 36

= \frac{π}{36}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

约分:

2

= \frac{πn}{3}

= \frac{π}{36} + \frac{πn}{3}

x > \frac{π}{36} + \frac{πn}{3}

x > \frac{π}{36} + \frac{πn}{3}

x > \frac{π}{36} + \frac{πn}{3}

6 x < π - arcsin (\frac{1}{2}) + 2 πn : x < \frac{5 π}{36} + \frac{π}{3} n

6 x < π - arcsin (\frac{1}{2}) + 2 πn

化简

π - arcsin (\frac{1}{2}) + 2 πn : π - \frac{π}{6} + 2 πn

π - arcsin (\frac{1}{2}) + 2 πn

使用以下普通恒等式:

arcsin (\frac{1}{2}) = \frac{π}{6}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - \frac{π}{6} + 2 πn

6 x < π - \frac{π}{6} + 2 πn

两边除以

6

6 x < π - \frac{π}{6} + 2 πn

两边除以

6

\frac{6 x}{6} < \frac{π}{6} - \frac{\frac{π}{6}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} < \frac{π}{6} - \frac{\frac{π}{6}}{6} + \frac{2 πn}{6}

化简

\frac{6 x}{6} : x

\frac{6 x}{6}

数字相除:

\frac{6}{6} = 1

= x

化简

\frac{π}{6} - \frac{\frac{π}{6}}{6} + \frac{2 πn}{6} : \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

\frac{π}{6} - \frac{\frac{π}{6}}{6} + \frac{2 πn}{6}

\frac{\frac{π}{6}}{6} = \frac{π}{36}

\frac{\frac{π}{6}}{6}

使用分式法则:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 6}

数字相乘:

6 \cdot 6 = 36

= \frac{π}{36}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

约分:

2

= \frac{πn}{3}

= \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

x < \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

x < \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

化简

\frac{π}{6} - \frac{π}{36} : \frac{5 π}{36}

\frac{π}{6} - \frac{π}{36}

6, 36

的最小公倍数:

36

6, 36

最小公倍数 (LCM)

6

质因数分解:

2 \cdot 3

6

6

除以

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 3

36

质因数分解:

2 \cdot 2 \cdot 3 \cdot 3

36

36

除以

236 = 18 \cdot 2

= 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3 \cdot 3

将每个因子乘以它在

6

或

36

中出现的最多次数

= 2 \cdot 2 \cdot 3 \cdot 3

数字相乘:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

根据最小公倍数调整分式

将每个分子乘以其分母转变为最小公倍数所要乘以的同一数值

36

对于

\frac{π}{6} :

将分母和分子乘以

6

\frac{π}{6} = \frac{π 6}{6 \cdot 6} = \frac{π 6}{36}

= \frac{π 6}{36} - \frac{π}{36}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π 6 - π}{36}

同类项相加:

6 π - π = 5 π

= \frac{5 π}{36}

x < \frac{5 π}{36} + \frac{π}{3} n

x < \frac{5 π}{36} + \frac{π}{3} n

合并区间

x > \frac{π}{36} + \frac{πn}{3} and x < \frac{5 π}{36} + \frac{π}{3} n

合并重叠的区间

\frac{π}{36} + \frac{π}{3} n < x < \frac{5 π}{36} + \frac{π}{3} n

流行的例子

cos (- θ) < 0

1 + cos (x) > 0

cos(2x)<-(sqrt(2))/2 ,0<= x<= 2pi

cos (2 x) < - \frac{2}{2}, 0 \leq x \leq 2 π

cos^2(x)-sin^2(x)-cos(x)<= 0

cos^{2} (x) - sin^{2} (x) - cos (x) \leq 0

sin(x/6)>=-(sqrt(2))/2

sin (\frac{x}{6}) \geq - \frac{2}{2}