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Popular Trigonometry >

cos(3x-pi/6)>0

Solution

cos (3 x - \frac{π}{6}) > 0

Solution

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

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

Solution steps

cos (3 x - \frac{π}{6}) > 0

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

- arccos (0) + 2 πn < (3 x - \frac{π}{6}) < arccos (0) + 2 πn

a < u < b

then

a < u and u < b

- arccos (0) + 2 πn < 3 x - \frac{π}{6} and 3 x - \frac{π}{6} < arccos (0) + 2 πn

- arccos (0) + 2 πn < 3 x - \frac{π}{6} : x > \frac{2 πn}{3} - \frac{π}{9}

- arccos (0) + 2 πn < 3 x - \frac{π}{6}

Switch sides

3 x - \frac{π}{6} > - arccos (0) + 2 πn

Simplify

- arccos (0) + 2 πn : - \frac{π}{2} + 2 πn

- arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

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}{2} + 2 πn

3 x - \frac{π}{6} > - \frac{π}{2} + 2 πn

Move

\frac{π}{6}

to the right side

3 x - \frac{π}{6} > - \frac{π}{2} + 2 πn

Add

\frac{π}{6}

to both sides

3 x - \frac{π}{6} + \frac{π}{6} > - \frac{π}{2} + 2 πn + \frac{π}{6}

Simplify

3 x - \frac{π}{6} + \frac{π}{6} > - \frac{π}{2} + 2 πn + \frac{π}{6}

Simplify

3 x - \frac{π}{6} + \frac{π}{6} : 3 x

3 x - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} > 0

= 3 x

Simplify

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

- \frac{π}{2} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn - \frac{π}{2} + \frac{π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

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

2

6

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

= - \frac{π 3}{6} + \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + π}{6}

Add similar elements:

- 3 π + π = - 2 π

= \frac{- 2 π}{6}

Apply the fraction rule:

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

= - \frac{2 π}{6}

Cancel the common factor:

2

= 2 πn - \frac{π}{3}

3 x > 2 πn - \frac{π}{3}

3 x > 2 πn - \frac{π}{3}

3 x > 2 πn - \frac{π}{3}

Divide both sides by

3

3 x > 2 πn - \frac{π}{3}

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

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

x > \frac{2 πn}{3} - \frac{π}{9}

x > \frac{2 πn}{3} - \frac{π}{9}

x > \frac{2 πn}{3} - \frac{π}{9}

3 x - \frac{π}{6} < arccos (0) + 2 πn : x < \frac{2 πn}{3} + \frac{2 π}{9}

3 x - \frac{π}{6} < arccos (0) + 2 πn

Simplify

arccos (0) + 2 πn : \frac{π}{2} + 2 πn

arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

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}{2} + 2 πn

3 x - \frac{π}{6} < \frac{π}{2} + 2 πn

Move

\frac{π}{6}

to the right side

3 x - \frac{π}{6} < \frac{π}{2} + 2 πn

Add

\frac{π}{6}

to both sides

3 x - \frac{π}{6} + \frac{π}{6} < \frac{π}{2} + 2 πn + \frac{π}{6}

Simplify

3 x - \frac{π}{6} + \frac{π}{6} < \frac{π}{2} + 2 πn + \frac{π}{6}

Simplify

3 x - \frac{π}{6} + \frac{π}{6} : 3 x

3 x - \frac{π}{6} + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} < 0

= 3 x

Simplify

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

\frac{π}{2} + 2 πn + \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{2} + \frac{π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

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

2

6

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

= \frac{π 3}{6} + \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π}{6}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{6}

Cancel the common factor:

2

= 2 πn + \frac{2 π}{3}

3 x < 2 πn + \frac{2 π}{3}

3 x < 2 πn + \frac{2 π}{3}

3 x < 2 πn + \frac{2 π}{3}

Divide both sides by

3

3 x < 2 πn + \frac{2 π}{3}

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

x < \frac{2 πn}{3} + \frac{2 π}{9}

x < \frac{2 πn}{3} + \frac{2 π}{9}

x < \frac{2 πn}{3} + \frac{2 π}{9}

Combine the intervals

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

Merge Overlapping Intervals

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

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