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Popular Trigonometry >

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

Solution

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

Solution

4 πn \leq x < \frac{π}{2} + 4 πn or \frac{3 π}{2} + 4 πn < x < \frac{5 π}{2} + 4 πn or \frac{7 π}{2} + 4 πn < x < 4 π + 4 πn

Interval Notation

[4 πn, \frac{π}{2} + 4 πn) \cup (\frac{3 π}{2} + 4 πn, \frac{5 π}{2} + 4 πn) \cup (\frac{7 π}{2} + 4 πn, 4 π + 4 πn)

Decimal

4 πn \leq x < 1.57079 \dots + 4 πn or 4.71238 \dots + 4 πn < x < 7.85398 \dots + 4 πn or 10.99557 \dots + 4 πn < x < 12.56637 \dots + 4 πn

Solution steps

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

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

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

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

Simplify

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

arcsin (- \frac{2}{2}) + 2 πn

arcsin (- \frac{2}{2}) = - \frac{π}{4}

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

Use the following property:

arcsin (- x) = - arcsin (x)

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

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

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{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}{4}

= - \frac{π}{4}

= - \frac{π}{4} + 2 πn

\frac{x}{2} > - \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} > - \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} > - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} > - 2 \cdot \frac{π}{4} + 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}{4} + 2 \cdot 2 πn : - \frac{π}{2} + 4 πn

- 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= - \frac{π}{2} + 4 πn

x > - \frac{π}{2} + 4 πn

x > - \frac{π}{2} + 4 πn

x > - \frac{π}{2} + 4 πn

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

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

Simplify

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

π - arcsin (- \frac{2}{2}) + 2 πn

arcsin (- \frac{2}{2}) = - \frac{π}{4}

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

Use the following property:

arcsin (- x) = - arcsin (x)

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

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

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{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}{4}

= - \frac{π}{4}

= π - (- \frac{π}{4}) + 2 πn

Apply rule

- (- a) = a

= π + \frac{π}{4} + 2 πn

\frac{x}{2} < π + \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} < π + \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} < 2 π + 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} < 2 π + 2 \cdot \frac{π}{4} + 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}{4} + 2 \cdot 2 πn : 2 π + \frac{π}{2} + 4 πn

2 π + 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 2 π + \frac{π}{2} + 4 πn

x < 2 π + \frac{π}{2} + 4 πn

x < 2 π + \frac{π}{2} + 4 πn

Simplify

2 π + \frac{π}{2} : \frac{5 π}{2}

2 π + \frac{π}{2}

Convert element to fraction:

2 π = \frac{2 π 2}{2}

= \frac{2 π 2}{2} + \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 2 + π}{2}

2 π 2 + π = 5 π

2 π 2 + π

Multiply the numbers:

2 \cdot 2 = 4

= 4 π + π

Add similar elements:

4 π + π = 5 π

= 5 π

= \frac{5 π}{2}

x < \frac{5 π}{2} + 4 πn

x < \frac{5 π}{2} + 4 πn

Combine the intervals

x > - \frac{π}{2} + 4 πn and x < \frac{5 π}{2} + 4 πn

Merge Overlapping Intervals

- \frac{π}{2} + 4 πn < x < \frac{5 π}{2} + 4 πn

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

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

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

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

Simplify

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

- π - arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

\frac{x}{2} > - π - \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} > - π - \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} > - 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} > - 2 π - 2 \cdot \frac{π}{4} + 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}{4} + 2 \cdot 2 πn : - 2 π - \frac{π}{2} + 4 πn

- 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= - 2 π - \frac{π}{2} + 4 πn

x > - 2 π - \frac{π}{2} + 4 πn

x > - 2 π - \frac{π}{2} + 4 πn

Simplify

- 2 π - \frac{π}{2} : - \frac{5 π}{2}

- 2 π - \frac{π}{2}

Convert element to fraction:

2 π = \frac{2 π 2}{2}

= - \frac{2 π 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 π 2 - π}{2}

- 2 π 2 - π = - 5 π

- 2 π 2 - π

Multiply the numbers:

2 \cdot 2 = 4

= - 4 π - π

Add similar elements:

- 4 π - π = - 5 π

= - 5 π

= \frac{- 5 π}{2}

Apply the fraction rule:

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

= - \frac{5 π}{2}

x > - \frac{5 π}{2} + 4 πn

x > - \frac{5 π}{2} + 4 πn

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

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

Simplify

arcsin (\frac{2}{2}) + 2 πn : \frac{π}{4} + 2 πn

arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

\frac{x}{2} < \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{x}{2} < \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} < 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} < 2 \cdot \frac{π}{4} + 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}{4} + 2 \cdot 2 πn : \frac{π}{2} + 4 πn

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x < \frac{π}{2} + 4 πn

x < \frac{π}{2} + 4 πn

x < \frac{π}{2} + 4 πn

Combine the intervals

x > - \frac{5 π}{2} + 4 πn and x < \frac{π}{2} + 4 πn

Merge Overlapping Intervals

- \frac{5 π}{2} + 4 πn < x < \frac{π}{2} + 4 πn

Combine the intervals

- \frac{π}{2} + 4 πn < x < \frac{5 π}{2} + 4 πn and - \frac{5 π}{2} + 4 πn < x < \frac{π}{2} + 4 πn

Merge Overlapping Intervals

4 πn \leq x < \frac{π}{2} + 4 πn or \frac{3 π}{2} + 4 πn < x < \frac{5 π}{2} + 4 πn or \frac{7 π}{2} + 4 πn < x < 4 π + 4 πn

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