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受欢迎的三角函数 >

4*sin^2(X)-2<0

解答

4 \cdot sin^{2} (X) - 2 < 0

解答

2 πn \leq X < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < X < \frac{5 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < X < 2 π + 2 πn

+2

间隔符号

[2 πn, \frac{π}{4} + 2 πn) \cup (\frac{3 π}{4} + 2 πn, \frac{5 π}{4} + 2 πn) \cup (\frac{7 π}{4} + 2 πn, 2 π + 2 πn)

十进制

2 πn \leq X < 0.78539 \dots + 2 πn or 2.35619 \dots + 2 πn < X < 3.92699 \dots + 2 πn or 5.49778 \dots + 2 πn < X < 6.28318 \dots + 2 πn

求解步骤

4 sin^{2} (X) - 2 < 0

将

2

到右边

4 sin^{2} (X) - 2 < 0

两边加上

2

4 sin^{2} (X) - 2 + 2 < 0 + 2

化简

4 sin^{2} (X) < 2

4 sin^{2} (X) < 2

两边除以

4

4 sin^{2} (X) < 2

两边除以

4

\frac{4 sin ^{2} ( X )}{4} < \frac{2}{4}

化简

sin^{2} (X) < \frac{1}{2}

sin^{2} (X) < \frac{1}{2}

对于

u^{n} < a

，若

n

为偶数，则

- n a < u < n a

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

若

a < u < b

，则

a < u and u < b

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

- \frac{1}{2} < sin (X) : - \frac{π}{4} + 2 πn < X < \frac{5 π}{4} + 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}{4}

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}{4}

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}{4}

= - \frac{π}{4}

化简

π - arcsin (- \frac{1}{2}) : \frac{5 π}{4}

π - arcsin (- \frac{1}{2})

arcsin (- \frac{1}{2}) = - \frac{π}{4}

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}{4}

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}{4}

= - \frac{π}{4}

= π - (- \frac{π}{4})

化简

π - (- \frac{π}{4})

使用法则

- (- a) = a

= π + \frac{π}{4}

将项转换为分式:

π = \frac{π 4}{4}

= \frac{π 4}{4} + \frac{π}{4}

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 4 + π}{4}

同类项相加:

4 π + π = 5 π

= \frac{5 π}{4}

= \frac{5 π}{4}

- \frac{π}{4} + 2 πn < X < \frac{5 π}{4} + 2 πn

sin (X) < \frac{1}{2} : - \frac{5 π}{4} + 2 πn < X < \frac{π}{4} + 2 πn

sin (X) < \frac{1}{2}

对于

sin (x) < a

，若

- 1 < a \leq 1

，则

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

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

化简

- π - arcsin (\frac{1}{2}) : - \frac{5 π}{4}

- π - arcsin (\frac{1}{2})

使用以下普通恒等式:

arcsin (\frac{1}{2}) = \frac{π}{4}

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}{4}

化简

- π - \frac{π}{4}

将项转换为分式:

π = \frac{π 4}{4}

= - \frac{π 4}{4} - \frac{π}{4}

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 4 - π}{4}

同类项相加:

- 4 π - π = - 5 π

= \frac{- 5 π}{4}

使用分式法则:

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

= - \frac{5 π}{4}

= - \frac{5 π}{4}

化简

arcsin (\frac{1}{2}) : \frac{π}{4}

arcsin (\frac{1}{2})

使用以下普通恒等式:

arcsin (\frac{1}{2}) = \frac{π}{4}

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}{4}

- \frac{5 π}{4} + 2 πn < X < \frac{π}{4} + 2 πn

合并区间

- \frac{π}{4} + 2 πn < X < \frac{5 π}{4} + 2 πn and - \frac{5 π}{4} + 2 πn < X < \frac{π}{4} + 2 πn

合并重叠的区间

2 πn \leq X < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < X < \frac{5 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < X < 2 π + 2 πn

流行的例子

6 cos (2 x) < 0

-(sqrt(3))/2 sin(x)-1/2 cos(x)<= 0

- \frac{3}{2} sin (x) - \frac{1}{2} cos (x) \leq 0

3/2 cos(x)-2>= cos(x)-7/4

\frac{3}{2} cos (x) - 2 \geq cos (x) - \frac{7}{4}

sin (x) < \frac{2}{π}

2 cos (x) \leq 1