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Popular >

-1<sin(x/6)<1

Solution

- 1 < sin (\frac{x}{6}) < 1

Solution

12 πn \leq x < 3 π + 12 πn or 3 π + 12 πn < x < 9 π + 12 πn or 9 π + 12 πn < x < 12 π + 12 πn

Interval Notation

[12 πn, 3 π + 12 πn) \cup (3 π + 12 πn, 9 π + 12 πn) \cup (9 π + 12 πn, 12 π + 12 πn)

Decimal Notation

12 πn \leq x < 9.42477 \dots + 12 πn or 9.42477 \dots + 12 πn < x < 28.27433 \dots + 12 πn or 28.27433 \dots + 12 πn < x < 37.69911 \dots + 12 πn

Solution steps

- 1 < sin (\frac{x}{6}) < 1

a < u < b

then

a < u and u < b

- 1 < sin (\frac{x}{6}) and sin (\frac{x}{6}) < 1

- 1 < sin (\frac{x}{6}) : - 3 π + 12 πn < x < 9 π + 12 πn

- 1 < sin (\frac{x}{6})

Switch sides

sin (\frac{x}{6}) > - 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 < \frac{x}{6} < π - arcsin (- 1) + 2 πn

a < u < b

then

a < u and u < b

arcsin (- 1) + 2 πn < \frac{x}{6} and \frac{x}{6} < π - arcsin (- 1) + 2 πn

arcsin (- 1) + 2 πn < \frac{x}{6} : x > - 3 π + 12 πn

arcsin (- 1) + 2 πn < \frac{x}{6}

Switch sides

\frac{x}{6} > 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

\frac{x}{6} > - \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{x}{6} > - \frac{π}{2} + 2 πn

Multiply both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

- 6 \cdot \frac{π}{2} + 6 \cdot 2 πn : - 3 π + 12 πn

- 6 \cdot \frac{π}{2} + 6 \cdot 2 πn

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= - 3 π + 12 πn

x > - 3 π + 12 πn

x > - 3 π + 12 πn

x > - 3 π + 12 πn

\frac{x}{6} < π - arcsin (- 1) + 2 πn : x < 9 π + 12 πn

\frac{x}{6} < 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

\frac{x}{6} < π + \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{x}{6} < π + \frac{π}{2} + 2 πn

Multiply both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

6 π + 6 \cdot \frac{π}{2} + 6 \cdot 2 πn : 9 π + 12 πn

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

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 6 π + 3 π + 12 πn

Add similar elements:

6 π + 3 π = 9 π

= 9 π + 12 πn

x < 9 π + 12 πn

x < 9 π + 12 πn

x < 9 π + 12 πn

Combine the intervals

x > - 3 π + 12 πn and x < 9 π + 12 πn

Merge Overlapping Intervals

- 3 π + 12 πn < x < 9 π + 12 πn

sin (\frac{x}{6}) < 1 : - 9 π + 12 πn < x < 3 π + 12 πn

sin (\frac{x}{6}) < 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 < \frac{x}{6} < arcsin (1) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (1) + 2 πn < \frac{x}{6} and \frac{x}{6} < arcsin (1) + 2 πn

- π - arcsin (1) + 2 πn < \frac{x}{6} : x > - 9 π + 12 πn

- π - arcsin (1) + 2 πn < \frac{x}{6}

Switch sides

\frac{x}{6} > - 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

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

Multiply both sides by

6

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

Multiply both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

- 6 π - 6 \cdot \frac{π}{2} + 6 \cdot 2 πn : - 9 π + 12 πn

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

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= - 6 π - 3 π + 12 πn

Add similar elements:

- 6 π - 3 π = - 9 π

= - 9 π + 12 πn

x > - 9 π + 12 πn

x > - 9 π + 12 πn

x > - 9 π + 12 πn

\frac{x}{6} < arcsin (1) + 2 πn : x < 3 π + 12 πn

\frac{x}{6} < 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

\frac{x}{6} < \frac{π}{2} + 2 πn

Multiply both sides by

6

\frac{x}{6} < \frac{π}{2} + 2 πn

Multiply both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

6 \cdot \frac{π}{2} + 6 \cdot 2 πn : 3 π + 12 πn

6 \cdot \frac{π}{2} + 6 \cdot 2 πn

6 \cdot \frac{π}{2} = 3 π

6 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 3 π + 12 πn

x < 3 π + 12 πn

x < 3 π + 12 πn

x < 3 π + 12 πn

Combine the intervals

x > - 9 π + 12 πn and x < 3 π + 12 πn

Merge Overlapping Intervals

- 9 π + 12 πn < x < 3 π + 12 πn

Combine the intervals

- 3 π + 12 πn < x < 9 π + 12 πn and - 9 π + 12 πn < x < 3 π + 12 πn

Merge Overlapping Intervals

12 πn \leq x < 3 π + 12 πn or 3 π + 12 πn < x < 9 π + 12 πn or 9 π + 12 πn < x < 12 π + 12 πn

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