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Popular >

sin(-x)> 1/2

Solution

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

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

Interval Notation

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

Decimal Notation

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

Solution steps

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

Use the following identity:

sin (- x) = - sin (x)

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

Multiply both sides by

- 1

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

Multiply both sides by -1 (reverse the inequality)

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

Simplify

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

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

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

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

Simplify

- 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})

Use the following property:

arcsin (- x) = - arcsin (x)

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

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

Use the following trivial identity:

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})

Simplify

- π - (- \frac{π}{6})

Apply rule

- (- a) = a

= - π + \frac{π}{6}

Convert element to fraction:

π = \frac{π 6}{6}

= - \frac{π 6}{6} + \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 6 + π}{6}

Add similar elements:

- 6 π + π = - 5 π

= \frac{- 5 π}{6}

Apply the fraction rule:

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

= - \frac{5 π}{6}

= - \frac{5 π}{6}

Simplify

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

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

Use the following property:

arcsin (- x) = - arcsin (x)

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

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

Use the following trivial identity:

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

sin (x + \frac{π}{5}) > \frac{1}{2}

cos (\frac{x}{2}) \leq 0

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

3 cos (4 x) + sin (4 x) > 2

sin (5 x) > 5, 0 \leq x \leq 2 π