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Popular Trigonometry >

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

Solution

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

Solution

\frac{π}{12} + \frac{2 π}{3} n \leq x \leq \frac{π}{4} + \frac{2 π}{3} n

Interval Notation

[\frac{π}{12} + \frac{2 π}{3} n, \frac{π}{4} + \frac{2 π}{3} n]

Decimal

0.26179 \dots + \frac{2 π}{3} n \leq x \leq 0.78539 \dots + \frac{2 π}{3} n

Solution steps

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

Move

\frac{2}{2}

to the right side

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

Add

\frac{2}{2}

to both sides

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

Simplify

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

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

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

arcsin (\frac{2}{2}) + 2 πn \leq 3 x

Switch sides

3 x \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

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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} + 2 πn

3 x \geq \frac{π}{4} + 2 πn

Divide both sides by

3

3 x \geq \frac{π}{4} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{π}{12}

= \frac{π}{12} + \frac{2 πn}{3}

x \geq \frac{π}{12} + \frac{2 πn}{3}

x \geq \frac{π}{12} + \frac{2 πn}{3}

x \geq \frac{π}{12} + \frac{2 πn}{3}

3 x \leq π - arcsin (\frac{2}{2}) + 2 πn : x \leq \frac{π}{4} + \frac{2 π}{3} n

3 x \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

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

3 x \leq π - \frac{π}{4} + 2 πn

Divide both sides by

3

3 x \leq π - \frac{π}{4} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{π}{12}

= \frac{π}{3} - \frac{π}{12} + \frac{2 πn}{3}

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

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

Simplify

\frac{π}{3} - \frac{π}{12} : \frac{π}{4}

\frac{π}{3} - \frac{π}{12}

Least Common Multiplier of

3, 12 : 12

3, 12

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

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

3

12

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 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}{3} :

multiply the denominator and numerator by

4

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

= \frac{π 4}{12} - \frac{π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 - π}{12}

Add similar elements:

4 π - π = 3 π

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

x \leq \frac{π}{4} + \frac{2 π}{3} n

x \leq \frac{π}{4} + \frac{2 π}{3} n

Combine the intervals

x \geq \frac{π}{12} + \frac{2 πn}{3} and x \leq \frac{π}{4} + \frac{2 π}{3} n

Merge Overlapping Intervals

\frac{π}{12} + \frac{2 π}{3} n \leq x \leq \frac{π}{4} + \frac{2 π}{3} n

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