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Popular >

-1<sin(pix)<1

Solution

- 1 < sin (π x) < 1

Solution

2 n \leq x < \frac{1}{2} + 2 n or \frac{1}{2} + 2 n < x < \frac{3}{2} + 2 n or \frac{3}{2} + 2 n < x < 2 + 2 n

Interval Notation

[2 n, \frac{1}{2} + 2 n) \cup (\frac{1}{2} + 2 n, \frac{3}{2} + 2 n) \cup (\frac{3}{2} + 2 n, 2 + 2 n)

Decimal Notation

2 n \leq x < 0.5 + 2 n or 0.5 + 2 n < x < 1.5 + 2 n or 1.5 + 2 n < x < 2 + 2 n

Solution steps

- 1 < sin (π x) < 1

a < u < b

then

a < u and u < b

- 1 < sin (π x) and sin (π x) < 1

- 1 < sin (π x) : - \frac{1}{2} + 2 n < x < \frac{3}{2} + 2 n

- 1 < sin (π x)

Switch sides

sin (π x) > - 1

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (- 1) + 2 πn < π x < π - arcsin (- 1) + 2 πn

a < u < b

then

a < u and u < b

arcsin (- 1) + 2 πn < π x and π x < π - arcsin (- 1) + 2 πn

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

arcsin (- 1) + 2 πn < π x

Switch sides

π x > arcsin (- 1) + 2 πn

Simplify

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

arcsin (- 1) + 2 πn

arcsin (- 1) = - \frac{π}{2}

arcsin (- 1)

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 1) = - arcsin (1)

= - arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{2}

arcsin (1)

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}{2}

= - \frac{π}{2}

= - \frac{π}{2} + 2 πn

π x > - \frac{π}{2} + 2 πn

Divide both sides by

π

π x > - \frac{π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

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

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

\frac{\frac{π}{2}}{π} = \frac{1}{2}

\frac{\frac{π}{2}}{π}

Apply the fraction rule:

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

= \frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

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

x > - \frac{1}{2} + 2 n

x > - \frac{1}{2} + 2 n

x > - \frac{1}{2} + 2 n

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

π x < π - arcsin (- 1) + 2 πn

Simplify

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

π - arcsin (- 1) + 2 πn

arcsin (- 1) = - \frac{π}{2}

arcsin (- 1)

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 1) = - arcsin (1)

= - arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{2}

arcsin (1)

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}{2}

= - \frac{π}{2}

= π - (- \frac{π}{2}) + 2 πn

Apply rule

- (- a) = a

= π + \frac{π}{2} + 2 πn

π x < π + \frac{π}{2} + 2 πn

Divide both sides by

π

π x < π + \frac{π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{π}{π} + \frac{\frac{π}{2}}{π} + \frac{2 πn}{π} : 1 + \frac{1}{2} + 2 n

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

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{\frac{π}{2}}{π} + \frac{2 πn}{π}

\frac{\frac{π}{2}}{π} = \frac{1}{2}

\frac{\frac{π}{2}}{π}

Apply the fraction rule:

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

= \frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= 1 + \frac{1}{2} + 2 n

x < 1 + \frac{1}{2} + 2 n

x < 1 + \frac{1}{2} + 2 n

Simplify

1 + \frac{1}{2} : \frac{3}{2}

1 + \frac{1}{2}

Convert element to fraction:

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

= \frac{1 \cdot 2}{2} + \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 2 + 1}{2}

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 + 1

Add the numbers:

2 + 1 = 3

= 3

= \frac{3}{2}

x < \frac{3}{2} + 2 n

x < \frac{3}{2} + 2 n

Combine the intervals

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

Merge Overlapping Intervals

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

sin (π x) < 1 : - \frac{3}{2} + 2 n < x < \frac{1}{2} + 2 n

sin (π x) < 1

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (1) + 2 πn < π x < arcsin (1) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (1) + 2 πn < π x and π x < arcsin (1) + 2 πn

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

- π - arcsin (1) + 2 πn < π x

Switch sides

π x > - π - arcsin (1) + 2 πn

Simplify

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

- π - arcsin (1) + 2 πn

Use the following trivial identity:

arcsin (1) = \frac{π}{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}

= - π - \frac{π}{2} + 2 πn

π x > - π - \frac{π}{2} + 2 πn

Divide both sides by

π

π x > - π - \frac{π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

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

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

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= - 1 - \frac{\frac{π}{2}}{π} + \frac{2 πn}{π}

\frac{\frac{π}{2}}{π} = \frac{1}{2}

\frac{\frac{π}{2}}{π}

Apply the fraction rule:

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

= \frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

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

x > - 1 - \frac{1}{2} + 2 n

x > - 1 - \frac{1}{2} + 2 n

Simplify

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

- 1 - \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

- 1 \cdot 2 - 1 = - 3

- 1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= - 2 - 1

Subtract the numbers:

- 2 - 1 = - 3

= - 3

= \frac{- 3}{2}

Apply the fraction rule:

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

= - \frac{3}{2}

x > - \frac{3}{2} + 2 n

x > - \frac{3}{2} + 2 n

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

π x < arcsin (1) + 2 πn

Simplify

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

arcsin (1) + 2 πn

Use the following trivial identity:

arcsin (1) = \frac{π}{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}{2} + 2 πn

π x < \frac{π}{2} + 2 πn

Divide both sides by

π

π x < \frac{π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{π}{2}}{π} + \frac{2 πn}{π} : \frac{1}{2} + 2 n

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

\frac{\frac{π}{2}}{π} = \frac{1}{2}

\frac{\frac{π}{2}}{π}

Apply the fraction rule:

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

= \frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{1}{2} + 2 n

x < \frac{1}{2} + 2 n

x < \frac{1}{2} + 2 n

x < \frac{1}{2} + 2 n

Combine the intervals

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

Merge Overlapping Intervals

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

Combine the intervals

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

Merge Overlapping Intervals

2 n \leq x < \frac{1}{2} + 2 n or \frac{1}{2} + 2 n < x < \frac{3}{2} + 2 n or \frac{3}{2} + 2 n < x < 2 + 2 n

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