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受欢迎的三角函数 >

cos^2(x/3)<= sin^2(x/3)-1/2

解答

cos^{2} (\frac{x}{3}) \leq sin^{2} (\frac{x}{3}) - \frac{1}{2}

解答

π + 3 πn \leq x \leq 2 π + 3 πn

+2

间隔符号

[π + 3 πn, 2 π + 3 πn]

十进制

3.14159 \dots + 3 πn \leq x \leq 6.28318 \dots + 3 πn

求解步骤

cos^{2} (\frac{x}{3}) \leq sin^{2} (\frac{x}{3}) - \frac{1}{2}

将

sin^{2} (\frac{x}{3})

para o lado esquerdo

cos^{2} (\frac{x}{3}) \leq sin^{2} (\frac{x}{3}) - \frac{1}{2}

两边减去

sin^{2} (\frac{x}{3})

cos^{2} (\frac{x}{3}) - sin^{2} (\frac{x}{3}) \leq sin^{2} (\frac{x}{3}) - \frac{1}{2} - sin^{2} (\frac{x}{3})

cos^{2} (\frac{x}{3}) - sin^{2} (\frac{x}{3}) \leq - \frac{1}{2}

cos^{2} (\frac{x}{3}) - sin^{2} (\frac{x}{3}) \leq - \frac{1}{2}

利用以下特性:

cos^{2} (x) - sin^{2} (x) = cos (2 x)

cos (2 \cdot \frac{x}{3}) \leq - \frac{1}{2}

化简

cos (2 \cdot \frac{x}{3}) : cos (\frac{2 x}{3})

cos (2 \cdot \frac{x}{3})

乘

2 \cdot \frac{x}{3} : \frac{2 x}{3}

2 \cdot \frac{x}{3}

分式相乘:

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

= \frac{x \cdot 2}{3}

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

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

cos (\frac{2 x}{3}) \leq - \frac{1}{2}

对于

cos (x) \leq a

，若

- 1 < a < 1

，则

arccos (a) + 2 πn \leq x \leq 2 π - arccos (a) + 2 πn

arccos (- \frac{1}{2}) + 2 πn \leq \frac{2 x}{3} \leq 2 π - arccos (- \frac{1}{2}) + 2 πn

若

a \leq u \leq b

，则

a \leq u and u \leq b

arccos (- \frac{1}{2}) + 2 πn \leq \frac{2 x}{3} and \frac{2 x}{3} \leq 2 π - arccos (- \frac{1}{2}) + 2 πn

arccos (- \frac{1}{2}) + 2 πn \leq \frac{2 x}{3} : x \geq π + 3 πn

arccos (- \frac{1}{2}) + 2 πn \leq \frac{2 x}{3}

交换两边

\frac{2 x}{3} \geq arccos (- \frac{1}{2}) + 2 πn

化简

arccos (- \frac{1}{2}) + 2 πn : \frac{2 π}{3} + 2 πn

arccos (- \frac{1}{2}) + 2 πn

使用以下普通恒等式:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{2 π}{3} + 2 πn

\frac{2 x}{3} \geq \frac{2 π}{3} + 2 πn

在两边乘以

3

\frac{2 x}{3} \geq \frac{2 π}{3} + 2 πn

在两边乘以

3

\frac{3 \cdot 2 x}{3} \geq 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

化简

\frac{3 \cdot 2 x}{3} \geq 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

化简

\frac{3 \cdot 2 x}{3} : 2 x

\frac{3 \cdot 2 x}{3}

数字相乘:

3 \cdot 2 = 6

= \frac{6 x}{3}

数字相除:

\frac{6}{3} = 2

= 2 x

化简

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

3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

分式相乘:

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

= \frac{2 π 3}{3}

约分:

3

= 2 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

数字相乘:

3 \cdot 2 = 6

= 6 πn

= 2 π + 6 πn

2 x \geq 2 π + 6 πn

2 x \geq 2 π + 6 πn

2 x \geq 2 π + 6 πn

两边除以

2

2 x \geq 2 π + 6 πn

两边除以

2

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

化简

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

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

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

\frac{2 π}{2} + \frac{6 πn}{2}

数字相除:

\frac{2}{2} = 1

= π + \frac{6 πn}{2}

数字相除:

\frac{6}{2} = 3

= π + 3 πn

x \geq π + 3 πn

x \geq π + 3 πn

x \geq π + 3 πn

\frac{2 x}{3} \leq 2 π - arccos (- \frac{1}{2}) + 2 πn : x \leq 2 π + 3 πn

\frac{2 x}{3} \leq 2 π - arccos (- \frac{1}{2}) + 2 πn

化简

2 π - arccos (- \frac{1}{2}) + 2 πn : 2 π - \frac{2 π}{3} + 2 πn

2 π - arccos (- \frac{1}{2}) + 2 πn

使用以下普通恒等式:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{2 π}{3} + 2 πn

\frac{2 x}{3} \leq 2 π - \frac{2 π}{3} + 2 πn

在两边乘以

3

\frac{2 x}{3} \leq 2 π - \frac{2 π}{3} + 2 πn

在两边乘以

3

\frac{3 \cdot 2 x}{3} \leq 3 \cdot 2 π - 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

化简

\frac{3 \cdot 2 x}{3} \leq 3 \cdot 2 π - 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

化简

\frac{3 \cdot 2 x}{3} : 2 x

\frac{3 \cdot 2 x}{3}

数字相乘:

3 \cdot 2 = 6

= \frac{6 x}{3}

数字相除:

\frac{6}{3} = 2

= 2 x

化简

3 \cdot 2 π - 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn : 4 π + 6 πn

3 \cdot 2 π - 3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

3 \cdot 2 π = 6 π

3 \cdot 2 π

数字相乘:

3 \cdot 2 = 6

= 6 π

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

分式相乘:

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

= \frac{2 π 3}{3}

约分:

3

= 2 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

数字相乘:

3 \cdot 2 = 6

= 6 πn

= 6 π - 2 π + 6 πn

同类项相加:

6 π - 2 π = 4 π

= 4 π + 6 πn

2 x \leq 4 π + 6 πn

2 x \leq 4 π + 6 πn

2 x \leq 4 π + 6 πn

两边除以

2

2 x \leq 4 π + 6 πn

两边除以

2

\frac{2 x}{2} \leq \frac{4 π}{2} + \frac{6 πn}{2}

化简

\frac{2 x}{2} \leq \frac{4 π}{2} + \frac{6 πn}{2}

化简

\frac{2 x}{2} : x

\frac{2 x}{2}

数字相除:

\frac{2}{2} = 1

= x

化简

\frac{4 π}{2} + \frac{6 πn}{2} : 2 π + 3 πn

\frac{4 π}{2} + \frac{6 πn}{2}

数字相除:

\frac{4}{2} = 2

= 2 π + \frac{6 πn}{2}

数字相除:

\frac{6}{2} = 3

= 2 π + 3 πn

x \leq 2 π + 3 πn

x \leq 2 π + 3 πn

x \leq 2 π + 3 πn

合并区间

x \geq π + 3 πn and x \leq 2 π + 3 πn

合并重叠的区间

π + 3 πn \leq x \leq 2 π + 3 πn

流行的例子

pi/2-arctan(e^x)>0.01

\frac{π}{2} - arctan (e^{x}) > 0.01

(cos (x) - 1) \geq 0

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

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

(sin(x)+4)(sqrt(3)-2sin(x))>0

(sin (x) + 4) (3 - 2 sin (x)) > 0

6.5<= 5+3sin(30t)

6.5 \leq 5 + 3 sin (30 t)