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sin(3x)cos(3x)-1/4 >= 0

Solution

sin (3 x) cos (3 x) - \frac{1}{4} \geq 0

Solution

\frac{π}{36} + \frac{π}{3} n \leq x \leq \frac{5 π}{36} + \frac{π}{3} n

Interval Notation

[\frac{π}{36} + \frac{π}{3} n, \frac{5 π}{36} + \frac{π}{3} n]

Decimal Notation

0.08726 \dots + \frac{π}{3} n \leq x \leq 0.43633 \dots + \frac{π}{3} n

Solution steps

sin (3 x) cos (3 x) - \frac{1}{4} \geq 0

Use the following identity:

2 cos (x) sin (x) = sin (2 x)

Therefore

cos (x) sin (x) = \frac{sin ( 2 x )}{2}

- \frac{1}{4} + \frac{sin ( 2 \cdot 3 x )}{2} \geq 0

Simplify

- \frac{1}{4} + \frac{sin ( 2 \cdot 3 x )}{2} : - \frac{1}{4} + \frac{1}{2} sin (6 x)

- \frac{1}{4} + \frac{sin ( 2 \cdot 3 x )}{2}

Multiply the numbers:

2 \cdot 3 = 6

= - \frac{1}{4} + \frac{sin ( 6 x )}{2}

= - \frac{1}{4} + \frac{1}{2} sin (6 x)

- \frac{1}{4} + \frac{1}{2} sin (6 x) \geq 0

Move

\frac{1}{4}

to the right side

- \frac{1}{4} + \frac{1}{2} sin (6 x) \geq 0

Add

\frac{1}{4}

to both sides

- \frac{1}{4} + \frac{1}{2} sin (6 x) + \frac{1}{4} \geq 0 + \frac{1}{4}

Simplify

\frac{1}{2} sin (6 x) \geq \frac{1}{4}

\frac{1}{2} sin (6 x) \geq \frac{1}{4}

Multiply both sides by

2

\frac{1}{2} sin (6 x) \geq \frac{1}{4}

Multiply both sides by

2

2 \cdot \frac{1}{2} sin (6 x) \geq \frac{1 \cdot 2}{4}

Simplify

2 \cdot \frac{1}{2} sin (6 x) \geq \frac{1 \cdot 2}{4}

Simplify

2 \cdot \frac{1}{2} sin (6 x) : sin (6 x)

2 \cdot \frac{1}{2} sin (6 x)

Multiply fractions:

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

= \frac{1 \cdot 2}{2} sin (6 x)

Cancel the common factor:

2

= sin (6 x) \cdot 1

Multiply:

sin (6 x) \cdot 1 = sin (6 x)

= sin (6 x)

Simplify

\frac{1 \cdot 2}{4} : \frac{1}{2}

\frac{1 \cdot 2}{4}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4}

Cancel the common factor:

2

= \frac{1}{2}

sin (6 x) \geq \frac{1}{2}

sin (6 x) \geq \frac{1}{2}

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

a \leq u \leq b

then

a \leq u and u \leq b

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

arcsin (\frac{1}{2}) + 2 πn \leq 6 x : x \geq \frac{π}{36} + \frac{πn}{3}

arcsin (\frac{1}{2}) + 2 πn \leq 6 x

Switch sides

6 x \geq arcsin (\frac{1}{2}) + 2 πn

Simplify

arcsin (\frac{1}{2}) + 2 πn : \frac{π}{6} + 2 πn

arcsin (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}{6} + 2 πn

6 x \geq \frac{π}{6} + 2 πn

Divide both sides by

6

6 x \geq \frac{π}{6} + 2 πn

Divide both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{\frac{π}{6}}{6} + \frac{2 πn}{6} : \frac{π}{36} + \frac{πn}{3}

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

\frac{\frac{π}{6}}{6} = \frac{π}{36}

\frac{\frac{π}{6}}{6}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 6}

Multiply the numbers:

6 \cdot 6 = 36

= \frac{π}{36}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

Cancel the common factor:

2

= \frac{πn}{3}

= \frac{π}{36} + \frac{πn}{3}

x \geq \frac{π}{36} + \frac{πn}{3}

x \geq \frac{π}{36} + \frac{πn}{3}

x \geq \frac{π}{36} + \frac{πn}{3}

6 x \leq π - arcsin (\frac{1}{2}) + 2 πn : x \leq \frac{5 π}{36} + \frac{π}{3} n

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

Simplify

π - arcsin (\frac{1}{2}) + 2 πn : π - \frac{π}{6} + 2 πn

π - arcsin (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}{6} + 2 πn

6 x \leq π - \frac{π}{6} + 2 πn

Divide both sides by

6

6 x \leq π - \frac{π}{6} + 2 πn

Divide both sides by

6

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

Simplify

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

Simplify

\frac{6 x}{6} : x

\frac{6 x}{6}

Divide the numbers:

\frac{6}{6} = 1

= x

Simplify

\frac{π}{6} - \frac{\frac{π}{6}}{6} + \frac{2 πn}{6} : \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

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

\frac{\frac{π}{6}}{6} = \frac{π}{36}

\frac{\frac{π}{6}}{6}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 6}

Multiply the numbers:

6 \cdot 6 = 36

= \frac{π}{36}

\frac{2 πn}{6} = \frac{πn}{3}

\frac{2 πn}{6}

Cancel the common factor:

2

= \frac{πn}{3}

= \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

x \leq \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

x \leq \frac{π}{6} - \frac{π}{36} + \frac{πn}{3}

Simplify

\frac{π}{6} - \frac{π}{36} : \frac{5 π}{36}

\frac{π}{6} - \frac{π}{36}

Least Common Multiplier of

6, 36 : 36

6, 36

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

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

6

36

= 2 \cdot 2 \cdot 3 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 \cdot 3 = 36

= 36

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

36

For

\frac{π}{6} :

multiply the denominator and numerator by

6

\frac{π}{6} = \frac{π 6}{6 \cdot 6} = \frac{π 6}{36}

= \frac{π 6}{36} - \frac{π}{36}

Since the denominators are equal, combine the fractions:

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

= \frac{π 6 - π}{36}

Add similar elements:

6 π - π = 5 π

= \frac{5 π}{36}

x \leq \frac{5 π}{36} + \frac{π}{3} n

x \leq \frac{5 π}{36} + \frac{π}{3} n

Combine the intervals

x \geq \frac{π}{36} + \frac{πn}{3} and x \leq \frac{5 π}{36} + \frac{π}{3} n

Merge Overlapping Intervals

\frac{π}{36} + \frac{π}{3} n \leq x \leq \frac{5 π}{36} + \frac{π}{3} n

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