English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2sin(x/2)-1>0

Solution

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

Solution

\frac{π}{3} + 4 πn < x < \frac{5 π}{3} + 4 πn

Interval Notation

(\frac{π}{3} + 4 πn, \frac{5 π}{3} + 4 πn)

Decimal Notation

1.04719 \dots + 4 πn < x < 5.23598 \dots + 4 πn

Solution steps

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

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

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

a < u < b

then

a < u and u < b

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

arcsin (\frac{1}{2}) + 2 πn < \frac{x}{2} : x > \frac{π}{3} + 4 πn

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

Switch sides

\frac{x}{2} > arcsin (\frac{1}{2}) + 2 πn

Simplify

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

arcsin (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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} + 2 πn

\frac{x}{2} > \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} > \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} > 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} > 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

x > \frac{π}{3} + 4 πn

x > \frac{π}{3} + 4 πn

x > \frac{π}{3} + 4 πn

\frac{x}{2} < π - arcsin (\frac{1}{2}) + 2 πn : x < \frac{5 π}{3} + 4 πn

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

Simplify

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

π - arcsin (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}{6} + 2 πn

\frac{x}{2} < π - \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} < π - \frac{π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 π - 2 \cdot \frac{π}{6} + 2 \cdot 2 πn : 2 π - \frac{π}{3} + 4 πn

2 π - 2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 2 π - \frac{π}{3} + 4 πn

x < 2 π - \frac{π}{3} + 4 πn

x < 2 π - \frac{π}{3} + 4 πn

Simplify

2 π - \frac{π}{3} : \frac{5 π}{3}

2 π - \frac{π}{3}

Convert element to fraction:

2 π = \frac{2 π 3}{3}

= \frac{2 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{2 π 3 - π}{3}

2 π 3 - π = 5 π

2 π 3 - π

Multiply the numbers:

2 \cdot 3 = 6

= 6 π - π

Add similar elements:

6 π - π = 5 π

= 5 π

= \frac{5 π}{3}

x < \frac{5 π}{3} + 4 πn

x < \frac{5 π}{3} + 4 πn

Combine the intervals

x > \frac{π}{3} + 4 πn and x < \frac{5 π}{3} + 4 πn

Merge Overlapping Intervals

\frac{π}{3} + 4 πn < x < \frac{5 π}{3} + 4 πn

Popular Examples

cos(θ)>= 0

cos (θ) \geq 0

cos(2x+30)> 1/2

cos (2 x + 3 0^{\circ}) > \frac{1}{2}

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

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

cos(θ)> 1/2

cos (θ) > \frac{1}{2}

sin(3x)<(sqrt(2))/2

sin (3 x) < \frac{2}{2}