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受欢迎的三角函数 >

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

解答

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

解答

- \frac{5 π}{6} + 2 πn < x < - \frac{π}{6} + 2 πn

+2

间隔符号

(- \frac{5 π}{6} + 2 πn, - \frac{π}{6} + 2 πn)

十进制

- 2.61799 \dots + 2 πn < x < - 0.52359 \dots + 2 πn

求解步骤

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

利用以下特性:

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

因此

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

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

化简

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

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

乘开

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

- sin (x) (2 + sin (x))

使用分配律:

a (b + c) = ab + a c

a = - sin (x), b = 2, c = sin (x)

= - sin (x) \cdot 2 + (- sin (x)) sin (x)

使用加减运算法则

+ (- a) = - a

= - 2 sin (x) - sin (x) sin (x)

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

sin (x) sin (x)

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

数字相加:

1 + 1 = 2

= sin^{2} (x)

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

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

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

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

打开括号

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

使用加减运算法则

- (- a) = a, - (a) = - a

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

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

化简

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

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

对同类项分组

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

同类项相加:

- sin^{2} (x) + sin^{2} (x) = 0

= - 2 sin (x) - 1

= - 2 sin (x) - 1

- 2 sin (x) - 1 > 0

将

1

到右边

- 2 sin (x) - 1 > 0

两边加上

1

- 2 sin (x) - 1 + 1 > 0 + 1

化简

- 2 sin (x) > 1

- 2 sin (x) > 1

在两边乘以

- 1

- 2 sin (x) > 1

两边乘以 -1（不等式变号）

(- 2 sin (x)) (- 1) < 1 \cdot (- 1)

化简

2 sin (x) < - 1

2 sin (x) < - 1

两边除以

2

2 sin (x) < - 1

两边除以

2

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

化简

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

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}{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 = - 5 π

= \frac{- 5 π}{6}

使用分式法则:

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

= - \frac{5 π}{6}

= - \frac{5 π}{6}

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}

- \frac{5 π}{6} + 2 πn < x < - \frac{π}{6} + 2 πn

流行的例子

1/(sqrt(3))<tan(x)

\frac{1}{3} < tan (x)

6 cos (θ) \geq 0

arctan(x^4)>0.0001

arctan (x^{4}) > 0.0001

cos(2x)>0,sin(x)>0

cos (2 x) > 0, sin (x) > 0

2 sin (2 x) \leq 0