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Popular >

sin(3x)-(sqrt(3))/2 >= 0

Solution

sin (3 x) - \frac{3}{2} \geq 0

Solution

\frac{π}{9} + \frac{2 π}{3} n \leq x \leq \frac{2 π}{9} + \frac{2 π}{3} n

Interval Notation

[\frac{π}{9} + \frac{2 π}{3} n, \frac{2 π}{9} + \frac{2 π}{3} n]

Decimal Notation

0.34906 \dots + \frac{2 π}{3} n \leq x \leq 0.69813 \dots + \frac{2 π}{3} n

Solution steps

sin (3 x) - \frac{3}{2} \geq 0

Move

\frac{3}{2}

to the right side

sin (3 x) - \frac{3}{2} \geq 0

Add

\frac{3}{2}

to both sides

sin (3 x) - \frac{3}{2} + \frac{3}{2} \geq 0 + \frac{3}{2}

Simplify

sin (3 x) \geq \frac{3}{2}

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

a \leq u \leq b

then

a \leq u and u \leq b

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

arcsin (\frac{3}{2}) + 2 πn \leq 3 x : x \geq \frac{π}{9} + \frac{2 πn}{3}

arcsin (\frac{3}{2}) + 2 πn \leq 3 x

Switch sides

3 x \geq arcsin (\frac{3}{2}) + 2 πn

Simplify

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

arcsin (\frac{3}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

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}{3} + 2 πn

3 x \geq \frac{π}{3} + 2 πn

Divide both sides by

3

3 x \geq \frac{π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{3}}{3} + \frac{2 πn}{3} : \frac{π}{9} + \frac{2 πn}{3}

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{9} + \frac{2 πn}{3}

x \geq \frac{π}{9} + \frac{2 πn}{3}

x \geq \frac{π}{9} + \frac{2 πn}{3}

x \geq \frac{π}{9} + \frac{2 πn}{3}

3 x \leq π - arcsin (\frac{3}{2}) + 2 πn : x \leq \frac{2 π}{9} + \frac{2 π}{3} n

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

Simplify

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

π - arcsin (\frac{3}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

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}{3} + 2 πn

3 x \leq π - \frac{π}{3} + 2 πn

Divide both sides by

3

3 x \leq π - \frac{π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{3} - \frac{\frac{π}{3}}{3} + \frac{2 πn}{3} : \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3}

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3}

x \leq \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3}

x \leq \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3}

Simplify

\frac{π}{3} - \frac{π}{9} : \frac{2 π}{9}

\frac{π}{3} - \frac{π}{9}

Least Common Multiplier of

3, 9 : 9

3, 9

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

9 : 3 \cdot 3

9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3

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

3

9

= 3 \cdot 3

Multiply the numbers:

3 \cdot 3 = 9

= 9

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

9

For

\frac{π}{3} :

multiply the denominator and numerator by

3

\frac{π}{3} = \frac{π 3}{3 \cdot 3} = \frac{π 3}{9}

= \frac{π 3}{9} - \frac{π}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π}{9}

Add similar elements:

3 π - π = 2 π

= \frac{2 π}{9}

x \leq \frac{2 π}{9} + \frac{2 π}{3} n

x \leq \frac{2 π}{9} + \frac{2 π}{3} n

Combine the intervals

x \geq \frac{π}{9} + \frac{2 πn}{3} and x \leq \frac{2 π}{9} + \frac{2 π}{3} n

Merge Overlapping Intervals

\frac{π}{9} + \frac{2 π}{3} n \leq x \leq \frac{2 π}{9} + \frac{2 π}{3} n

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