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Popular Trigonometry >

sin(x/6)>=-(sqrt(2))/2

Solution

sin (\frac{x}{6}) \geq - \frac{2}{2}

Solution

- \frac{3 π}{2} + 12 πn \leq x \leq \frac{15 π}{2} + 12 πn

Interval Notation

[- \frac{3 π}{2} + 12 πn, \frac{15 π}{2} + 12 πn]

Decimal

- 4.71238 \dots + 12 πn \leq x \leq 23.56194 \dots + 12 πn

Solution steps

sin (\frac{x}{6}) \geq - \frac{2}{2}

For

sin (x) \geq a

, if

- 1 < a < 1

then

arcsin (a) + 2 πn \leq x \leq π - arcsin (a) + 2 πn

arcsin (- \frac{2}{2}) + 2 πn \leq \frac{x}{6} \leq π - arcsin (- \frac{2}{2}) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

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

arcsin (- \frac{2}{2}) + 2 πn \leq \frac{x}{6} : x \geq - \frac{3 π}{2} + 12 πn

arcsin (- \frac{2}{2}) + 2 πn \leq \frac{x}{6}

Switch sides

\frac{x}{6} \geq 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}{6} \geq - \frac{π}{4} + 2 πn

Multiply both sides by

6

\frac{x}{6} \geq - \frac{π}{4} + 2 πn

Multiply both sides by

6

\frac{6 x}{6} \geq - 6 \cdot \frac{π}{4} + 6 \cdot 2 πn

Simplify

\frac{6 x}{6} \geq - 6 \cdot \frac{π}{4} + 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}{4} + 6 \cdot 2 πn : - \frac{3 π}{2} + 12 πn

- 6 \cdot \frac{π}{4} + 6 \cdot 2 πn

6 \cdot \frac{π}{4} = \frac{3 π}{2}

6 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 6}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= - \frac{3 π}{2} + 12 πn

x \geq - \frac{3 π}{2} + 12 πn

x \geq - \frac{3 π}{2} + 12 πn

x \geq - \frac{3 π}{2} + 12 πn

\frac{x}{6} \leq π - arcsin (- \frac{2}{2}) + 2 πn : x \leq \frac{15 π}{2} + 12 πn

\frac{x}{6} \leq 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}{6} \leq π + \frac{π}{4} + 2 πn

Multiply both sides by

6

\frac{x}{6} \leq π + \frac{π}{4} + 2 πn

Multiply both sides by

6

\frac{6 x}{6} \leq 6 π + 6 \cdot \frac{π}{4} + 6 \cdot 2 πn

Simplify

\frac{6 x}{6} \leq 6 π + 6 \cdot \frac{π}{4} + 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}{4} + 6 \cdot 2 πn : 6 π + \frac{3 π}{2} + 12 πn

6 π + 6 \cdot \frac{π}{4} + 6 \cdot 2 πn

6 \cdot \frac{π}{4} = \frac{3 π}{2}

6 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 6}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 6 π + \frac{3 π}{2} + 12 πn

x \leq 6 π + \frac{3 π}{2} + 12 πn

x \leq 6 π + \frac{3 π}{2} + 12 πn

Simplify

6 π + \frac{3 π}{2} : \frac{15 π}{2}

6 π + \frac{3 π}{2}

Convert element to fraction:

6 π = \frac{6 π 2}{2}

= \frac{6 π 2}{2} + \frac{3 π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{6 π 2 + 3 π}{2}

6 π 2 + 3 π = 15 π

6 π 2 + 3 π

Multiply the numbers:

6 \cdot 2 = 12

= 12 π + 3 π

Add similar elements:

12 π + 3 π = 15 π

= 15 π

= \frac{15 π}{2}

x \leq \frac{15 π}{2} + 12 πn

x \leq \frac{15 π}{2} + 12 πn

Combine the intervals

x \geq - \frac{3 π}{2} + 12 πn and x \leq \frac{15 π}{2} + 12 πn

Merge Overlapping Intervals

- \frac{3 π}{2} + 12 πn \leq x \leq \frac{15 π}{2} + 12 πn

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