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Popular >

sin(2x-(2pi)/3)<=-(sqrt(2))/2

Solution

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

Solution

- \frac{π}{24} + πn \leq x \leq \frac{5 π}{24} + πn

Interval Notation

[- \frac{π}{24} + πn, \frac{5 π}{24} + πn]

Decimal Notation

- 0.13089 \dots + πn \leq x \leq 0.65449 \dots + πn

Solution steps

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

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

a \leq u \leq b

then

a \leq u and u \leq b

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

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

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

Switch sides

2 x - \frac{2 π}{3} \geq - 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

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

Move

\frac{2 π}{3}

to the right side

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

Add

\frac{2 π}{3}

to both sides

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

Simplify

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

Simplify

2 x - \frac{2 π}{3} + \frac{2 π}{3} : 2 x

2 x - \frac{2 π}{3} + \frac{2 π}{3}

Add similar elements:

- \frac{2 π}{3} + \frac{2 π}{3} \geq 0

= 2 x

Simplify

- π + \frac{π}{4} + 2 πn + \frac{2 π}{3} : - π + 2 πn + \frac{11 π}{12}

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

Group like terms

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

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{2 π}{3} :

multiply the denominator and numerator by

4

\frac{2 π}{3} = \frac{2 π 4}{3 \cdot 4} = \frac{8 π}{12}

= \frac{π 3}{12} + \frac{8 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + 8 π}{12}

Add similar elements:

3 π + 8 π = 11 π

= - π + 2 πn + \frac{11 π}{12}

2 x \geq - π + 2 πn + \frac{11 π}{12}

2 x \geq - π + 2 πn + \frac{11 π}{12}

2 x \geq - π + 2 πn + \frac{11 π}{12}

Divide both sides by

2

2 x \geq - π + 2 πn + \frac{11 π}{12}

Divide both sides by

2

\frac{2 x}{2} \geq - \frac{π}{2} + \frac{2 πn}{2} + \frac{\frac{11 π}{12}}{2}

Simplify

\frac{2 x}{2} \geq - \frac{π}{2} + \frac{2 πn}{2} + \frac{\frac{11 π}{12}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{π}{2} + \frac{2 πn}{2} + \frac{\frac{11 π}{12}}{2} : πn - \frac{π}{2} + \frac{11 π}{24}

- \frac{π}{2} + \frac{2 πn}{2} + \frac{\frac{11 π}{12}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{11 π}{12}}{2} = \frac{11 π}{24}

\frac{\frac{11 π}{12}}{2}

Apply the fraction rule:

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

= \frac{11 π}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{11 π}{24}

= - \frac{π}{2} + πn + \frac{11 π}{24}

Group like terms

= πn - \frac{π}{2} + \frac{11 π}{24}

x \geq πn - \frac{π}{2} + \frac{11 π}{24}

x \geq πn - \frac{π}{2} + \frac{11 π}{24}

Simplify

- \frac{π}{2} + \frac{11 π}{24} : - \frac{π}{24}

- \frac{π}{2} + \frac{11 π}{24}

Least Common Multiplier of

2, 24 : 24

2, 24

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

24 : 2 \cdot 2 \cdot 2 \cdot 3

24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

24

= 2 \cdot 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 3 = 24

= 24

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

24

For

\frac{π}{2} :

multiply the denominator and numerator by

12

\frac{π}{2} = \frac{π 12}{2 \cdot 12} = \frac{π 12}{24}

= - \frac{π 12}{24} + \frac{11 π}{24}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 12 + 11 π}{24}

Add similar elements:

- 12 π + 11 π = - π

= \frac{- π}{24}

Apply the fraction rule:

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

= - \frac{π}{24}

x \geq - \frac{π}{24} + πn

x \geq - \frac{π}{24} + πn

2 x - \frac{2 π}{3} \leq arcsin (- \frac{2}{2}) + 2 πn : x \leq πn + \frac{5 π}{24}

2 x - \frac{2 π}{3} \leq 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

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

Move

\frac{2 π}{3}

to the right side

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

Add

\frac{2 π}{3}

to both sides

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

Simplify

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

Simplify

2 x - \frac{2 π}{3} + \frac{2 π}{3} : 2 x

2 x - \frac{2 π}{3} + \frac{2 π}{3}

Add similar elements:

- \frac{2 π}{3} + \frac{2 π}{3} \leq 0

= 2 x

Simplify

- \frac{π}{4} + 2 πn + \frac{2 π}{3} : 2 πn + \frac{5 π}{12}

- \frac{π}{4} + 2 πn + \frac{2 π}{3}

Group like terms

= 2 πn - \frac{π}{4} + \frac{2 π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{2 π}{3} :

multiply the denominator and numerator by

4

\frac{2 π}{3} = \frac{2 π 4}{3 \cdot 4} = \frac{8 π}{12}

= - \frac{π 3}{12} + \frac{8 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 8 π}{12}

Add similar elements:

- 3 π + 8 π = 5 π

= 2 πn + \frac{5 π}{12}

2 x \leq 2 πn + \frac{5 π}{12}

2 x \leq 2 πn + \frac{5 π}{12}

2 x \leq 2 πn + \frac{5 π}{12}

Divide both sides by

2

2 x \leq 2 πn + \frac{5 π}{12}

Divide both sides by

2

\frac{2 x}{2} \leq \frac{2 πn}{2} + \frac{\frac{5 π}{12}}{2}

Simplify

\frac{2 x}{2} \leq \frac{2 πn}{2} + \frac{\frac{5 π}{12}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{\frac{5 π}{12}}{2} : πn + \frac{5 π}{24}

\frac{2 πn}{2} + \frac{\frac{5 π}{12}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{5 π}{12}}{2} = \frac{5 π}{24}

\frac{\frac{5 π}{12}}{2}

Apply the fraction rule:

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

= \frac{5 π}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{5 π}{24}

= πn + \frac{5 π}{24}

x \leq πn + \frac{5 π}{24}

x \leq πn + \frac{5 π}{24}

x \leq πn + \frac{5 π}{24}

Combine the intervals

x \geq - \frac{π}{24} + πn and x \leq πn + \frac{5 π}{24}

Merge Overlapping Intervals

- \frac{π}{24} + πn \leq x \leq \frac{5 π}{24} + πn

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