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Popular Trigonometry >

1/3 cos(3x-pi/3)< 1/6

Solution

\frac{1}{3} cos (3 x - \frac{π}{3}) < \frac{1}{6}

Solution

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

Interval Notation

(\frac{2 π}{9} + \frac{2 π}{3} n, \frac{2 π}{3} + \frac{2 π}{3} n)

Decimal

0.69813 \dots + \frac{2 π}{3} n < x < 2.09439 \dots + \frac{2 π}{3} n

Solution steps

\frac{1}{3} cos (3 x - \frac{π}{3}) < \frac{1}{6}

Multiply both sides by

3

\frac{1}{3} cos (3 x - \frac{π}{3}) < \frac{1}{6}

Multiply both sides by

3

3 \cdot \frac{1}{3} cos (3 x - \frac{π}{3}) < \frac{1 \cdot 3}{6}

Simplify

3 \cdot \frac{1}{3} cos (3 x - \frac{π}{3}) < \frac{1 \cdot 3}{6}

Simplify

3 \cdot \frac{1}{3} cos (3 x - \frac{π}{3}) : cos (3 x - \frac{π}{3})

3 \cdot \frac{1}{3} cos (3 x - \frac{π}{3})

Multiply fractions:

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

= \frac{1 \cdot 3}{3} cos (3 x - \frac{π}{3})

Cancel the common factor:

3

= cos (3 x - \frac{π}{3}) \cdot 1

Multiply:

cos (3 x - \frac{π}{3}) \cdot 1 = cos (3 x - \frac{π}{3})

= cos (3 x - \frac{π}{3})

Simplify

\frac{1 \cdot 3}{6} : \frac{1}{2}

\frac{1 \cdot 3}{6}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{6}

Cancel the common factor:

3

= \frac{1}{2}

cos (3 x - \frac{π}{3}) < \frac{1}{2}

cos (3 x - \frac{π}{3}) < \frac{1}{2}

cos (3 x - \frac{π}{3}) < \frac{1}{2}

For

cos (x) < a

, if

- 1 < a \leq 1

then

arccos (a) + 2 πn < x < 2 π - arccos (a) + 2 πn

arccos (\frac{1}{2}) + 2 πn < (3 x - \frac{π}{3}) < 2 π - arccos (\frac{1}{2}) + 2 πn

a < u < b

then

a < u and u < b

arccos (\frac{1}{2}) + 2 πn < 3 x - \frac{π}{3} and 3 x - \frac{π}{3} < 2 π - arccos (\frac{1}{2}) + 2 πn

arccos (\frac{1}{2}) + 2 πn < 3 x - \frac{π}{3} : x > \frac{2 π}{9} + \frac{2 πn}{3}

arccos (\frac{1}{2}) + 2 πn < 3 x - \frac{π}{3}

Switch sides

3 x - \frac{π}{3} > arccos (\frac{1}{2}) + 2 πn

Simplify

arccos (\frac{1}{2}) + 2 πn : \frac{π}{3} + 2 πn

arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{3} + 2 πn

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{3} + \frac{π}{3} : 3 x

3 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} > 0

= 3 x

Simplify

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

\frac{π}{3} + 2 πn + \frac{π}{3}

Group like terms

= \frac{π}{3} + \frac{π}{3} + 2 πn

Combine the fractions

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

Apply rule

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

= \frac{π + π}{3}

Add similar elements:

π + π = 2 π

= \frac{2 π}{3}

= \frac{2 π}{3} + 2 πn

3 x > \frac{2 π}{3} + 2 πn

3 x > \frac{2 π}{3} + 2 πn

3 x > \frac{2 π}{3} + 2 πn

Divide both sides by

3

3 x > \frac{2 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{2 π}{3}}{3} = \frac{2 π}{9}

\frac{\frac{2 π}{3}}{3}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{2 π}{9}

= \frac{2 π}{9} + \frac{2 πn}{3}

x > \frac{2 π}{9} + \frac{2 πn}{3}

x > \frac{2 π}{9} + \frac{2 πn}{3}

x > \frac{2 π}{9} + \frac{2 πn}{3}

3 x - \frac{π}{3} < 2 π - arccos (\frac{1}{2}) + 2 πn : x < \frac{2 π}{3} + \frac{2 πn}{3}

3 x - \frac{π}{3} < 2 π - arccos (\frac{1}{2}) + 2 πn

Simplify

2 π - arccos (\frac{1}{2}) + 2 πn : 2 π - \frac{π}{3} + 2 πn

2 π - arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{3} + 2 πn

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{3} + \frac{π}{3} : 3 x

3 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} < 0

= 3 x

Simplify

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

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

Group like terms

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

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 2 π + 2 πn

3 x < 2 π + 2 πn

3 x < 2 π + 2 πn

3 x < 2 π + 2 πn

Divide both sides by

3

3 x < 2 π + 2 πn

Divide both sides by

3

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

Simplify

x < \frac{2 π}{3} + \frac{2 πn}{3}

x < \frac{2 π}{3} + \frac{2 πn}{3}

Combine the intervals

x > \frac{2 π}{9} + \frac{2 πn}{3} and x < \frac{2 π}{3} + \frac{2 πn}{3}

Merge Overlapping Intervals

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

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