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受欢迎的三角函数 >

sin(x)+cos(x)>1

解答

sin (x) + cos (x) > 1

解答

2 πn < x < \frac{π}{2} + 2 πn

+2

间隔符号

(2 πn, \frac{π}{2} + 2 πn)

十进制

2 πn < x < 1.57079 \dots + 2 πn

求解步骤

sin (x) + cos (x) > 1

利用以下特性:

cos (x) + sin (x) = 2 sin (\frac{π}{4} + x)

2 sin (\frac{π}{4} + x) > 1

两边除以

2

2 sin (\frac{π}{4} + x) > 1

两边除以

2

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

化简

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

化简

\frac{2 sin ( \frac{π}{4} + x )}{2} : sin (\frac{π}{4} + x)

\frac{2 sin ( \frac{π}{4} + x )}{2}

约分:

2

= sin (\frac{π}{4} + x)

化简

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

乘以共轭根式

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (\frac{π}{4} + x) > \frac{2}{2}

sin (\frac{π}{4} + x) > \frac{2}{2}

sin (\frac{π}{4} + x) > \frac{2}{2}

对于

sin (x) > a

，若

- 1 \leq a < 1

，则

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

arcsin (\frac{2}{2}) + 2 πn < (\frac{π}{4} + x) < π - arcsin (\frac{2}{2}) + 2 πn

若

a < u < b

，则

a < u and u < b

arcsin (\frac{2}{2}) + 2 πn < \frac{π}{4} + x and \frac{π}{4} + x < π - arcsin (\frac{2}{2}) + 2 πn

arcsin (\frac{2}{2}) + 2 πn < \frac{π}{4} + x : x > 2 πn

arcsin (\frac{2}{2}) + 2 πn < \frac{π}{4} + x

交换两边

\frac{π}{4} + x > arcsin (\frac{2}{2}) + 2 πn

化简

arcsin (\frac{2}{2}) + 2 πn : \frac{π}{4} + 2 πn

arcsin (\frac{2}{2}) + 2 πn

使用以下普通恒等式:

arcsin (\frac{2}{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} + 2 πn

\frac{π}{4} + x > \frac{π}{4} + 2 πn

将

\frac{π}{4}

到右边

\frac{π}{4} + x > \frac{π}{4} + 2 πn

两边减去

\frac{π}{4}

\frac{π}{4} + x - \frac{π}{4} > \frac{π}{4} + 2 πn - \frac{π}{4}

化简

\frac{π}{4} + x - \frac{π}{4} > \frac{π}{4} + 2 πn - \frac{π}{4}

化简

\frac{π}{4} + x - \frac{π}{4} : x

\frac{π}{4} + x - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} > 0

= x

化简

\frac{π}{4} + 2 πn - \frac{π}{4} : 2 πn

\frac{π}{4} + 2 πn - \frac{π}{4}

对同类项分组

= \frac{π}{4} - \frac{π}{4} + 2 πn

同类项相加:

\frac{π}{4} - \frac{π}{4} = 0

= 2 πn

x > 2 πn

x > 2 πn

x > 2 πn

\frac{π}{4} + x < π - arcsin (\frac{2}{2}) + 2 πn : x < \frac{π}{2} + 2 πn

\frac{π}{4} + x < π - arcsin (\frac{2}{2}) + 2 πn

化简

π - arcsin (\frac{2}{2}) + 2 πn : π - \frac{π}{4} + 2 πn

π - arcsin (\frac{2}{2}) + 2 πn

使用以下普通恒等式:

arcsin (\frac{2}{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} + 2 πn

\frac{π}{4} + x < π - \frac{π}{4} + 2 πn

将

\frac{π}{4}

到右边

\frac{π}{4} + x < π - \frac{π}{4} + 2 πn

两边减去

\frac{π}{4}

\frac{π}{4} + x - \frac{π}{4} < π - \frac{π}{4} + 2 πn - \frac{π}{4}

化简

\frac{π}{4} + x - \frac{π}{4} < π - \frac{π}{4} + 2 πn - \frac{π}{4}

化简

\frac{π}{4} + x - \frac{π}{4} : x

\frac{π}{4} + x - \frac{π}{4}

同类项相加:

\frac{π}{4} - \frac{π}{4} < 0

= x

化简

π - \frac{π}{4} + 2 πn - \frac{π}{4} : - \frac{π}{2} + π + 2 πn

π - \frac{π}{4} + 2 πn - \frac{π}{4}

对同类项分组

= - \frac{π}{4} - \frac{π}{4} + π + 2 πn

合并分式

- \frac{π}{4} - \frac{π}{4} : - \frac{π}{2}

使用法则

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

= \frac{- π - π}{4}

同类项相加:

- π - π = - 2 π

= \frac{- 2 π}{4}

使用分式法则:

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

= - \frac{2 π}{4}

约分:

2

= - \frac{π}{2}

= - \frac{π}{2} + π + 2 πn

x < - \frac{π}{2} + π + 2 πn

x < - \frac{π}{2} + π + 2 πn

x < - \frac{π}{2} + π + 2 πn

化简

- \frac{π}{2} + π : \frac{π}{2}

- \frac{π}{2} + π

将项转换为分式:

π = \frac{π 2}{2}

= - \frac{π}{2} + \frac{π 2}{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 + π 2}{2}

同类项相加:

- π + 2 π = π

= \frac{π}{2}

x < \frac{π}{2} + 2 πn

合并区间

x > 2 πn and x < \frac{π}{2} + 2 πn

合并重叠的区间

2 πn < x < \frac{π}{2} + 2 πn

流行的例子

sin(x)<-(sqrt(2))/2

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

cos(x)-sin(x)+1>= 0

cos (x) - sin (x) + 1 \geq 0

tan(x)>\sqrt[4]{5}

tan (x) > 45

3 tan (x) < - 3

-1/2 (10sin(2pi*60x))<-0.52

- \frac{1}{2} (10 sin (2 π \cdot 60 x)) < - 0.52