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受欢迎的三角函数 >

-1/2 <sin(x)<1

解答

- \frac{1}{2} < sin (x) < 1

解答

2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{7 π}{6} + 2 πn or \frac{11 π}{6} + 2 πn < x < 2 π + 2 πn

+2

间隔符号

[2 πn, \frac{π}{2} + 2 πn) \cup (\frac{π}{2} + 2 πn, \frac{7 π}{6} + 2 πn) \cup (\frac{11 π}{6} + 2 πn, 2 π + 2 πn)

十进制

2 πn \leq x < 1.57079 \dots + 2 πn or 1.57079 \dots + 2 πn < x < 3.66519 \dots + 2 πn or 5.75958 \dots + 2 πn < x < 6.28318 \dots + 2 πn

求解步骤

- \frac{1}{2} < sin (x) < 1

若

a < u < b

，则

a < u and u < b

- \frac{1}{2} < sin (x) and sin (x) < 1

- \frac{1}{2} < sin (x) : - \frac{π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn

- \frac{1}{2} < sin (x)

交换两边

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

化简

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

arcsin (- \frac{1}{2})

利用以下特性:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

使用以下普通恒等式:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

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}

= - \frac{π}{6}

化简

π - arcsin (- \frac{1}{2}) : \frac{7 π}{6}

π - arcsin (- \frac{1}{2})

arcsin (- \frac{1}{2}) = - \frac{π}{6}

arcsin (- \frac{1}{2})

利用以下特性:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

使用以下普通恒等式:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{2})

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}

= - \frac{π}{6}

= π - (- \frac{π}{6})

化简

π - (- \frac{π}{6})

使用法则

- (- a) = a

= π + \frac{π}{6}

将项转换为分式:

π = \frac{π 6}{6}

= \frac{π 6}{6} + \frac{π}{6}

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}{6}

同类项相加:

6 π + π = 7 π

= \frac{7 π}{6}

= \frac{7 π}{6}

- \frac{π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn

sin (x) < 1 : - \frac{3 π}{2} + 2 πn < x < \frac{π}{2} + 2 πn

sin (x) < 1

对于

sin (x) < a

，若

- 1 < a \leq 1

，则

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

- π - arcsin (1) + 2 πn < x < arcsin (1) + 2 πn

化简

- π - arcsin (1) : - \frac{3 π}{2}

- π - arcsin (1)

使用以下普通恒等式:

arcsin (1) = \frac{π}{2}

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}{2}

化简

- π - \frac{π}{2}

将项转换为分式:

π = \frac{π 2}{2}

= - \frac{π 2}{2} - \frac{π}{2}

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 2 - π}{2}

同类项相加:

- 2 π - π = - 3 π

= \frac{- 3 π}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{3 π}{2}

= - \frac{3 π}{2}

化简

arcsin (1) : \frac{π}{2}

arcsin (1)

使用以下普通恒等式:

arcsin (1) = \frac{π}{2}

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}{2}

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

合并区间

- \frac{π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn and - \frac{3 π}{2} + 2 πn < x < \frac{π}{2} + 2 πn

合并重叠的区间

2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{7 π}{6} + 2 πn or \frac{11 π}{6} + 2 πn < x < 2 π + 2 πn

流行的例子

0<= x<= 2piarccos(1/2)

0 \leq x \leq 2 π arccos (\frac{1}{2})

24sin(θ)>0.06>24cos(θ)

24 sin (θ) > 0.06 > 24 cos (θ)

cos(x)<= sin(x)<= (sqrt(3))/2

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

0<= θ<360sin(71)

0 \leq θ < 360 sin (7 1^{\circ})

1<= tan(x)<= 3^{1/2}

1 \leq tan (x) \leq 3^{\frac{1}{2}}