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Popular Trigonometry >

sin((pix)/2)> 1/2

Solution

sin (\frac{π x}{2}) > \frac{1}{2}

Solution

\frac{1}{3} + 4 n < x < \frac{5}{3} + 4 n

Interval Notation

(\frac{1}{3} + 4 n, \frac{5}{3} + 4 n)

Decimal

0.33333 \dots + 4 n < x < 1.66666 \dots + 4 n

Solution steps

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

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

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

Switch sides

\frac{π 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{π x}{2} > \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{π x}{2} > \frac{π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Divide the numbers:

\frac{2}{2} = 1

= π 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

π x > \frac{π}{3} + 4 πn

π x > \frac{π}{3} + 4 πn

π x > \frac{π}{3} + 4 πn

Divide both sides by

π

π x > \frac{π}{3} + 4 πn

Divide both sides by

π

\frac{π x}{π} > \frac{\frac{π}{3}}{π} + \frac{4 πn}{π}

Simplify

\frac{π x}{π} > \frac{\frac{π}{3}}{π} + \frac{4 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{π}{3}}{π} + \frac{4 πn}{π} : \frac{1}{3} + 4 n

\frac{\frac{π}{3}}{π} + \frac{4 πn}{π}

\frac{\frac{π}{3}}{π} = \frac{1}{3}

\frac{\frac{π}{3}}{π}

Apply the fraction rule:

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

= \frac{π}{3 π}

Cancel the common factor:

π

= \frac{1}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= \frac{1}{3} + 4 n

x > \frac{1}{3} + 4 n

x > \frac{1}{3} + 4 n

x > \frac{1}{3} + 4 n

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Divide the numbers:

\frac{2}{2} = 1

= π 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

π x < 2 π - \frac{π}{3} + 4 πn

π x < 2 π - \frac{π}{3} + 4 πn

π x < 2 π - \frac{π}{3} + 4 πn

Divide both sides by

π

π x < 2 π - \frac{π}{3} + 4 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{2 π}{π} - \frac{\frac{π}{3}}{π} + \frac{4 πn}{π} : 2 - \frac{1}{3} + 4 n

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

\frac{2 π}{π} = 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

\frac{\frac{π}{3}}{π} = \frac{1}{3}

\frac{\frac{π}{3}}{π}

Apply the fraction rule:

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

= \frac{π}{3 π}

Cancel the common factor:

π

= \frac{1}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= 2 - \frac{1}{3} + 4 n

x < 2 - \frac{1}{3} + 4 n

x < 2 - \frac{1}{3} + 4 n

Simplify

2 - \frac{1}{3} : \frac{5}{3}

2 - \frac{1}{3}

Convert element to fraction:

2 = \frac{2 \cdot 3}{3}

= \frac{2 \cdot 3}{3} - \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 3 - 1}{3}

2 \cdot 3 - 1 = 5

2 \cdot 3 - 1

Multiply the numbers:

2 \cdot 3 = 6

= 6 - 1

Subtract the numbers:

6 - 1 = 5

= 5

= \frac{5}{3}

x < \frac{5}{3} + 4 n

x < \frac{5}{3} + 4 n

Combine the intervals

x > \frac{1}{3} + 4 n and x < \frac{5}{3} + 4 n

Merge Overlapping Intervals

\frac{1}{3} + 4 n < x < \frac{5}{3} + 4 n

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