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受欢迎的三角函数 >

1-2cos(2x)>sin^2(x)

解答

1 - 2 cos (2 x) > sin^{2} (x)

解答

arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn or - π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn

+2

间隔符号

(arcsin (\frac{1}{3}) + 2 πn, π - arcsin (\frac{1}{3}) + 2 πn) \cup (- π + arcsin (\frac{1}{3}) + 2 πn, - arcsin (\frac{1}{3}) + 2 πn)

十进制

0.61547 \dots + 2 πn < x < 2.52611 \dots + 2 πn or - 2.52611 \dots + 2 πn < x < - 0.61547 \dots + 2 πn

求解步骤

1 - 2 cos (2 x) > sin^{2} (x)

将

sin^{2} (x)

para o lado esquerdo

1 - 2 cos (2 x) > sin^{2} (x)

两边减去

sin^{2} (x)

1 - 2 cos (2 x) - sin^{2} (x) > sin^{2} (x) - sin^{2} (x)

1 - 2 cos (2 x) - sin^{2} (x) > 0

1 - 2 cos (2 x) - sin^{2} (x) > 0

利用以下特性:

cos (2 x) = 1 - 2 sin^{2} (x)

1 - sin^{2} (x) - 2 (1 - 2 sin^{2} (x)) > 0

化简

1 - sin^{2} (x) - 2 (1 - 2 sin^{2} (x)) : 3 sin^{2} (x) - 1

1 - sin^{2} (x) - 2 (1 - 2 sin^{2} (x))

乘开

- 2 (1 - 2 sin^{2} (x)) : - 2 + 4 sin^{2} (x)

- 2 (1 - 2 sin^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = - 2, b = 1, c = 2 sin^{2} (x)

= - 2 \cdot 1 - (- 2) \cdot 2 sin^{2} (x)

使用加减运算法则

- (- a) = a

= - 2 \cdot 1 + 2 \cdot 2 sin^{2} (x)

化简

- 2 \cdot 1 + 2 \cdot 2 sin^{2} (x) : - 2 + 4 sin^{2} (x)

- 2 \cdot 1 + 2 \cdot 2 sin^{2} (x)

数字相乘:

2 \cdot 1 = 2

= - 2 + 2 \cdot 2 sin^{2} (x)

数字相乘:

2 \cdot 2 = 4

= - 2 + 4 sin^{2} (x)

= - 2 + 4 sin^{2} (x)

= 1 - sin^{2} (x) - 2 + 4 sin^{2} (x)

化简

1 - sin^{2} (x) - 2 + 4 sin^{2} (x) : 3 sin^{2} (x) - 1

1 - sin^{2} (x) - 2 + 4 sin^{2} (x)

对同类项分组

= - sin^{2} (x) + 4 sin^{2} (x) + 1 - 2

同类项相加:

- sin^{2} (x) + 4 sin^{2} (x) = 3 sin^{2} (x)

= 3 sin^{2} (x) + 1 - 2

数字相加/相减:

1 - 2 = - 1

= 3 sin^{2} (x) - 1

= 3 sin^{2} (x) - 1

3 sin^{2} (x) - 1 > 0

将

1

到右边

3 sin^{2} (x) - 1 > 0

两边加上

1

3 sin^{2} (x) - 1 + 1 > 0 + 1

化简

3 sin^{2} (x) > 1

3 sin^{2} (x) > 1

两边除以

3

3 sin^{2} (x) > 1

两边除以

3

\frac{3 sin ^{2} ( x )}{3} > \frac{1}{3}

化简

sin^{2} (x) > \frac{1}{3}

sin^{2} (x) > \frac{1}{3}

对于

u^{n} > a

，若

n

为偶数，则

u < - n a or u > n a

sin (x) < - \frac{1}{3} or sin (x) > \frac{1}{3}

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

sin (x) < - \frac{1}{3}

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}{3}) + 2 πn < x < arcsin (- \frac{1}{3}) + 2 πn

化简

- π - arcsin (- \frac{1}{3}) : - π + arcsin (\frac{1}{3})

- π - arcsin (- \frac{1}{3})

利用以下特性:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - π - (- arcsin (\frac{1}{3}))

使用法则

- (- a) = a

= - π + arcsin (\frac{1}{3})

化简

arcsin (- \frac{1}{3}) : - arcsin (\frac{1}{3})

arcsin (- \frac{1}{3})

利用以下特性:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - arcsin (\frac{1}{3})

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

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

sin (x) > \frac{1}{3}

对于

sin (x) > a

，若

- 1 \leq a < 1

，则

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

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

合并区间

- π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn or arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn

合并重叠的区间

arcsin (\frac{1}{3}) + 2 πn < x < π - arcsin (\frac{1}{3}) + 2 πn or - π + arcsin (\frac{1}{3}) + 2 πn < x < - arcsin (\frac{1}{3}) + 2 πn

流行的例子

cos^{2} (x) \geq \frac{1}{2}

cos^2(x)+(sqrt(2))/2 cos(x)>0

cos^{2} (x) + \frac{2}{2} cos (x) > 0

sin (x) \geq 0.5

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

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

sin (x) \geq - 1