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Popular >

2sin(x)+sqrt(3)>0

Solution

2 sin (x) + 3 > 0

- \frac{π}{3} + 2 πn < x < \frac{4 π}{3} + 2 πn

Interval Notation

(- \frac{π}{3} + 2 πn, \frac{4 π}{3} + 2 πn)

Decimal Notation

- 1.04719 \dots + 2 πn < x < 4.18879 \dots + 2 πn

Solution steps

2 sin (x) + 3 > 0

Move

3

to the right side

2 sin (x) + 3 > 0

Subtract

3

from both sides

2 sin (x) + 3 - 3 > 0 - 3

Simplify

2 sin (x) > - 3

2 sin (x) > - 3

Divide both sides by

2

2 sin (x) > - 3

Divide both sides by

2

\frac{2 sin ( x )}{2} > \frac{- 3}{2}

Simplify

sin (x) > - \frac{3}{2}

sin (x) > - \frac{3}{2}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

Simplify

arcsin (- \frac{3}{2}) : - \frac{π}{3}

arcsin (- \frac{3}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{2}) = - arcsin (\frac{3}{2})

= - arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

arcsin (\frac{3}{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}{3}

= - \frac{π}{3}

Simplify

π - arcsin (- \frac{3}{2}) : \frac{4 π}{3}

π - arcsin (- \frac{3}{2})

arcsin (- \frac{3}{2}) = - \frac{π}{3}

arcsin (- \frac{3}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{2}) = - arcsin (\frac{3}{2})

= - arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

arcsin (\frac{3}{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}{3}

= - \frac{π}{3}

= π - (- \frac{π}{3})

Simplify

π - (- \frac{π}{3})

Apply rule

- (- a) = a

= π + \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= \frac{π 3}{3} + \frac{π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π}{3}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{3}

= \frac{4 π}{3}

- \frac{π}{3} + 2 πn < x < \frac{4 π}{3} + 2 πn

cos^{2} (x) \geq 1

csc (θ) (\frac{26}{5}) tan (θ) < 0

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

sin (x) 0 < 2 π

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