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Popular Trigonometry >

3sin((pix)/(12)-pi/2)<=-2

Solution

3 sin (\frac{π x}{12} - \frac{π}{2}) \leq - 2

Solution

\frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π} + 24 n \leq x \leq \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π} + 24 n

Interval Notation

[\frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π} + 24 n, \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π} + 24 n]

Decimal

- 3.21264 \dots + 24 n \leq x \leq 3.21264 \dots + 24 n

Solution steps

3 sin (\frac{π x}{12} - \frac{π}{2}) \leq - 2

Divide both sides by

3

3 sin (\frac{π x}{12} - \frac{π}{2}) \leq - 2

Divide both sides by

3

\frac{3 sin ( \frac{π x}{12} - \frac{π}{2} )}{3} \leq \frac{- 2}{3}

Simplify

sin (\frac{π x}{12} - \frac{π}{2}) \leq - \frac{2}{3}

sin (\frac{π x}{12} - \frac{π}{2}) \leq - \frac{2}{3}

For

sin (x) \leq a

, if

- 1 < a < 1

then

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

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

a \leq u \leq b

then

a \leq u and u \leq b

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

- π - arcsin (- \frac{2}{3}) + 2 πn \leq \frac{π x}{12} - \frac{π}{2} : x \geq \frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π} + 24 n

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

Switch sides

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

Simplify

- π - arcsin (- \frac{2}{3}) + 2 πn : - π + arcsin (\frac{2}{3}) + 2 πn

- π - arcsin (- \frac{2}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{3}) = - arcsin (\frac{2}{3})

= - π - (- arcsin (\frac{2}{3})) + 2 πn

Apply rule

- (- a) = a

= - π + arcsin (\frac{2}{3}) + 2 πn

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

Move

\frac{π}{2}

to the right side

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

Add

\frac{π}{2}

to both sides

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

Simplify

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

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

Multiply both sides by

12

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

Multiply both sides by

12

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

Simplify

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

Simplify

\frac{12 π x}{12} : π x

\frac{12 π x}{12}

Divide the numbers:

\frac{12}{12} = 1

= π x

Simplify

- 12 π + 12 arcsin (\frac{2}{3}) + 12 \cdot 2 πn + 12 \cdot \frac{π}{2} : - 6 π + 24 πn + 12 arcsin (\frac{2}{3})

- 12 π + 12 arcsin (\frac{2}{3}) + 12 \cdot 2 πn + 12 \cdot \frac{π}{2}

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

12 \cdot \frac{π}{2} = 6 π

12 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 12}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

= - 12 π + 12 arcsin (\frac{2}{3}) + 24 πn + 6 π

Group like terms

= - 12 π + 6 π + 24 πn + 12 arcsin (\frac{2}{3})

Add similar elements:

- 12 π + 6 π = - 6 π

= - 6 π + 24 πn + 12 arcsin (\frac{2}{3})

π x \geq - 6 π + 24 πn + 12 arcsin (\frac{2}{3})

π x \geq - 6 π + 24 πn + 12 arcsin (\frac{2}{3})

π x \geq - 6 π + 24 πn + 12 arcsin (\frac{2}{3})

Divide both sides by

π

π x \geq - 6 π + 24 πn + 12 arcsin (\frac{2}{3})

Divide both sides by

π

\frac{π x}{π} \geq - \frac{6 π}{π} + \frac{24 πn}{π} + \frac{12 arcsin ( \frac{2}{3} )}{π}

Simplify

\frac{π x}{π} \geq - \frac{6 π}{π} + \frac{24 πn}{π} + \frac{12 arcsin ( \frac{2}{3} )}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{6 π}{π} + \frac{24 πn}{π} + \frac{12 arcsin ( \frac{2}{3} )}{π} : - 6 + 24 n + \frac{12 arcsin ( \frac{2}{3} )}{π}

- \frac{6 π}{π} + \frac{24 πn}{π} + \frac{12 arcsin ( \frac{2}{3} )}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= - 6 + \frac{24 πn}{π} + \frac{12 arcsin ( \frac{2}{3} )}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= - 6 + 24 n + \frac{12 arcsin ( \frac{2}{3} )}{π}

x \geq - 6 + 24 n + \frac{12 arcsin ( \frac{2}{3} )}{π}

x \geq - 6 + 24 n + \frac{12 arcsin ( \frac{2}{3} )}{π}

Simplify

- 6 + \frac{12 arcsin ( \frac{2}{3} )}{π} : \frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π}

- 6 + \frac{12 arcsin ( \frac{2}{3} )}{π}

Convert element to fraction:

6 = \frac{6 π}{π}

= - \frac{6 π}{π} + \frac{12 arcsin ( \frac{2}{3} )}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π}

x \geq \frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π} + 24 n

x \geq \frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π} + 24 n

\frac{π x}{12} - \frac{π}{2} \leq arcsin (- \frac{2}{3}) + 2 πn : x \leq \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π} + 24 n

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

Simplify

arcsin (- \frac{2}{3}) + 2 πn : - arcsin (\frac{2}{3}) + 2 πn

arcsin (- \frac{2}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{3}) = - arcsin (\frac{2}{3})

= - arcsin (\frac{2}{3}) + 2 πn

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

Move

\frac{π}{2}

to the right side

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

Add

\frac{π}{2}

to both sides

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

Simplify

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

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

Multiply both sides by

12

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

Multiply both sides by

12

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

Simplify

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

Simplify

\frac{12 π x}{12} : π x

\frac{12 π x}{12}

Divide the numbers:

\frac{12}{12} = 1

= π x

Simplify

- 12 arcsin (\frac{2}{3}) + 12 \cdot 2 πn + 12 \cdot \frac{π}{2} : - 12 arcsin (\frac{2}{3}) + 24 πn + 6 π

- 12 arcsin (\frac{2}{3}) + 12 \cdot 2 πn + 12 \cdot \frac{π}{2}

12 \cdot 2 πn = 24 πn

12 \cdot 2 πn

Multiply the numbers:

12 \cdot 2 = 24

= 24 πn

12 \cdot \frac{π}{2} = 6 π

12 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 12}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

= - 12 arcsin (\frac{2}{3}) + 24 πn + 6 π

π x \leq - 12 arcsin (\frac{2}{3}) + 24 πn + 6 π

π x \leq - 12 arcsin (\frac{2}{3}) + 24 πn + 6 π

π x \leq - 12 arcsin (\frac{2}{3}) + 24 πn + 6 π

Divide both sides by

π

π x \leq - 12 arcsin (\frac{2}{3}) + 24 πn + 6 π

Divide both sides by

π

\frac{π x}{π} \leq - \frac{12 arcsin ( \frac{2}{3} )}{π} + \frac{24 πn}{π} + \frac{6 π}{π}

Simplify

\frac{π x}{π} \leq - \frac{12 arcsin ( \frac{2}{3} )}{π} + \frac{24 πn}{π} + \frac{6 π}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{12 arcsin ( \frac{2}{3} )}{π} + \frac{24 πn}{π} + \frac{6 π}{π} : 6 + 24 n - \frac{12 arcsin ( \frac{2}{3} )}{π}

- \frac{12 arcsin ( \frac{2}{3} )}{π} + \frac{24 πn}{π} + \frac{6 π}{π}

Group like terms

= \frac{6 π}{π} + \frac{24 πn}{π} - \frac{12 arcsin ( \frac{2}{3} )}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= 6 + \frac{24 πn}{π} - \frac{12 arcsin ( \frac{2}{3} )}{π}

Cancel

\frac{24 πn}{π} : 24 n

\frac{24 πn}{π}

Cancel the common factor:

π

= 24 n

= 6 + 24 n - \frac{12 arcsin ( \frac{2}{3} )}{π}

x \leq 6 + 24 n - \frac{12 arcsin ( \frac{2}{3} )}{π}

x \leq 6 + 24 n - \frac{12 arcsin ( \frac{2}{3} )}{π}

Simplify

6 - \frac{12 arcsin ( \frac{2}{3} )}{π} : \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π}

6 - \frac{12 arcsin ( \frac{2}{3} )}{π}

Convert element to fraction:

6 = \frac{6 π}{π}

= \frac{6 π}{π} - \frac{12 arcsin ( \frac{2}{3} )}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π}

x \leq \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π} + 24 n

x \leq \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π} + 24 n

Combine the intervals

x \geq \frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π} + 24 n and x \leq \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π} + 24 n

Merge Overlapping Intervals

\frac{- 6 π + 12 arcsin ( \frac{2}{3} )}{π} + 24 n \leq x \leq \frac{6 π - 12 arcsin ( \frac{2}{3} )}{π} + 24 n

Popular Examples

-sin(x)(2+sin(x))-cos^2(x)>0