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Popular >

2sin(x/2)-1<0

Solution

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

Solution

- \frac{7 π}{3} + 4 πn < x < \frac{π}{3} + 4 πn

Interval Notation

(- \frac{7 π}{3} + 4 πn, \frac{π}{3} + 4 πn)

Decimal Notation

- 7.33038 \dots + 4 πn < x < 1.04719 \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 < a \leq 1

then

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

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

a < u < b

then

a < u and u < b

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

- π - arcsin (\frac{1}{2}) + 2 πn < \frac{x}{2} : x > - \frac{7 π}{3} + 4 πn

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

Switch sides

\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{7 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 = - 7 π

- 2 π 3 - π

Multiply the numbers:

2 \cdot 3 = 6

= - 6 π - π

Add similar elements:

- 6 π - π = - 7 π

= - 7 π

= \frac{- 7 π}{3}

Apply the fraction rule:

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

= - \frac{7 π}{3}

x > - \frac{7 π}{3} + 4 πn

x > - \frac{7 π}{3} + 4 πn

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

\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

Combine the intervals

x > - \frac{7 π}{3} + 4 πn and x < \frac{π}{3} + 4 πn

Merge Overlapping Intervals

- \frac{7 π}{3} + 4 πn < x < \frac{π}{3} + 4 πn

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