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Popular >

4cos^2(3x-1)>0

Solution

4 cos^{2} (3 x - 1) > 0

Solution

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

Interval Notation

(\frac{2 - π}{6} + \frac{2 π}{3} n, \frac{2 + π}{6} + \frac{2 π}{3} n) \cup (\frac{2 + π}{6} + \frac{2 π}{3} n, \frac{2 + 3 π}{6} + \frac{2 π}{3} n)

Decimal Notation

- 0.19026 \dots + \frac{2 π}{3} n < x < 0.85693 \dots + \frac{2 π}{3} n or 0.85693 \dots + \frac{2 π}{3} n < x < 1.90412 \dots + \frac{2 π}{3} n

Solution steps

4 cos^{2} (3 x - 1) > 0

Divide both sides by

4

4 cos^{2} (3 x - 1) > 0

Divide both sides by

4

\frac{4 cos ^{2} ( 3 x - 1 )}{4} > \frac{0}{4}

Simplify

cos^{2} (3 x - 1) > 0

cos^{2} (3 x - 1) > 0

For

u^{n} > 0

, if

n

is even then

u < 0

u > 0

cos (3 x - 1) < 0 or cos (3 x - 1) > 0

cos (3 x - 1) < 0 : \frac{2 + π}{6} + \frac{2 π}{3} n < x < \frac{2 + 3 π}{6} + \frac{2 π}{3} n

cos (3 x - 1) < 0

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

arccos (0) + 2 πn < (3 x - 1) < 2 π - arccos (0) + 2 πn

a < u < b

then

a < u and u < b

arccos (0) + 2 πn < 3 x - 1 and 3 x - 1 < 2 π - arccos (0) + 2 πn

arccos (0) + 2 πn < 3 x - 1 : x > \frac{2 + π}{6} + \frac{2 π}{3} n

arccos (0) + 2 πn < 3 x - 1

Switch sides

3 x - 1 > 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 - 1 > \frac{π}{2} + 2 πn

Move

1

to the right side

3 x - 1 > \frac{π}{2} + 2 πn

Add

1

to both sides

3 x - 1 + 1 > \frac{π}{2} + 2 πn + 1

Simplify

3 x > \frac{π}{2} + 2 πn + 1

3 x > \frac{π}{2} + 2 πn + 1

Divide both sides by

3

3 x > \frac{π}{2} + 2 πn + 1

Divide both sides by

3

\frac{3 x}{3} > \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Simplify

\frac{3 x}{3} > \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3} : \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Group like terms

= \frac{1}{3} + \frac{2 πn}{3} + \frac{\frac{π}{2}}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{1}{3} + \frac{2 πn}{3} + \frac{π}{6}

Group like terms

= \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

x > \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

x > \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

Simplify

\frac{1}{3} + \frac{π}{6} : \frac{2 + π}{6}

\frac{1}{3} + \frac{π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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{1}{3} :

multiply the denominator and numerator by

2

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

= \frac{2}{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{2 + π}{6}

x > \frac{2 + π}{6} + \frac{2 π}{3} n

x > \frac{2 + π}{6} + \frac{2 π}{3} n

3 x - 1 < 2 π - arccos (0) + 2 πn : x < \frac{2 + 3 π}{6} + \frac{2 π}{3} n

3 x - 1 < 2 π - arccos (0) + 2 πn

Simplify

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

2 π - 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}

= 2 π - \frac{π}{2} + 2 πn

3 x - 1 < 2 π - \frac{π}{2} + 2 πn

Move

1

to the right side

3 x - 1 < 2 π - \frac{π}{2} + 2 πn

Add

1

to both sides

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

Simplify

3 x < 2 π - \frac{π}{2} + 2 πn + 1

3 x < 2 π - \frac{π}{2} + 2 πn + 1

Divide both sides by

3

3 x < 2 π - \frac{π}{2} + 2 πn + 1

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 π}{3} - \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3} : \frac{1}{3} - \frac{π}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

\frac{2 π}{3} - \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Group like terms

= \frac{1}{3} + \frac{2 π}{3} + \frac{2 πn}{3} - \frac{\frac{π}{2}}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{1}{3} + \frac{2 π}{3} + \frac{2 πn}{3} - \frac{π}{6}

Group like terms

= \frac{1}{3} - \frac{π}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

x < \frac{1}{3} - \frac{π}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

x < \frac{1}{3} - \frac{π}{6} + \frac{2 π}{3} + \frac{2 πn}{3}

Simplify

\frac{1}{3} - \frac{π}{6} + \frac{2 π}{3} : \frac{2 + 3 π}{6}

\frac{1}{3} - \frac{π}{6} + \frac{2 π}{3}

Combine the fractions

\frac{1}{3} + \frac{2 π}{3} : \frac{1 + 2 π}{3}

Apply rule

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

= \frac{1 + 2 π}{3}

= \frac{2 π + 1}{3} - \frac{π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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{1 + 2 π}{3} :

multiply the denominator and numerator by

2

\frac{1 + 2 π}{3} = \frac{( 1 + 2 π ) \cdot 2}{3 \cdot 2} = \frac{( 1 + 2 π ) \cdot 2}{6}

= \frac{( 1 + 2 π ) \cdot 2}{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{( 1 + 2 π ) \cdot 2 - π}{6}

Expand

(1 + 2 π) \cdot 2 - π : 2 + 3 π

(1 + 2 π) \cdot 2 - π

= 2 (1 + 2 π) - π

Expand

2 (1 + 2 π) : 2 + 4 π

2 (1 + 2 π)

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = 1, c = 2 π

= 2 \cdot 1 + 2 \cdot 2 π

Simplify

2 \cdot 1 + 2 \cdot 2 π : 2 + 4 π

2 \cdot 1 + 2 \cdot 2 π

Multiply the numbers:

2 \cdot 1 = 2

= 2 + 2 \cdot 2 π

Multiply the numbers:

2 \cdot 2 = 4

= 2 + 4 π

= 2 + 4 π

= 2 + 4 π - π

Add similar elements:

4 π - π = 3 π

= 2 + 3 π

= \frac{2 + 3 π}{6}

x < \frac{2 + 3 π}{6} + \frac{2 π}{3} n

x < \frac{2 + 3 π}{6} + \frac{2 π}{3} n

x < \frac{2 + 3 π}{6} + \frac{2 π}{3} n

Combine the intervals

x > \frac{2 + π}{6} + \frac{2 π}{3} n and x < \frac{2 + 3 π}{6} + \frac{2 π}{3} n

Merge Overlapping Intervals

\frac{2 + π}{6} + \frac{2 π}{3} n < x < \frac{2 + 3 π}{6} + \frac{2 π}{3} n

cos (3 x - 1) > 0 : \frac{2 - π}{6} + \frac{2 π}{3} n < x < \frac{2 + π}{6} + \frac{2 π}{3} n

cos (3 x - 1) > 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 - 1) < arccos (0) + 2 πn

a < u < b

then

a < u and u < b

- arccos (0) + 2 πn < 3 x - 1 and 3 x - 1 < arccos (0) + 2 πn

- arccos (0) + 2 πn < 3 x - 1 : x > \frac{2 - π}{6} + \frac{2 π}{3} n

- arccos (0) + 2 πn < 3 x - 1

Switch sides

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

Move

1

to the right side

3 x - 1 > - \frac{π}{2} + 2 πn

Add

1

to both sides

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

Simplify

3 x > - \frac{π}{2} + 2 πn + 1

3 x > - \frac{π}{2} + 2 πn + 1

Divide both sides by

3

3 x > - \frac{π}{2} + 2 πn + 1

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

- \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3} : \frac{1}{3} - \frac{π}{6} + \frac{2 πn}{3}

- \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Group like terms

= \frac{1}{3} + \frac{2 πn}{3} - \frac{\frac{π}{2}}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{1}{3} + \frac{2 πn}{3} - \frac{π}{6}

Group like terms

= \frac{1}{3} - \frac{π}{6} + \frac{2 πn}{3}

x > \frac{1}{3} - \frac{π}{6} + \frac{2 πn}{3}

x > \frac{1}{3} - \frac{π}{6} + \frac{2 πn}{3}

Simplify

\frac{1}{3} - \frac{π}{6} : \frac{2 - π}{6}

\frac{1}{3} - \frac{π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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{1}{3} :

multiply the denominator and numerator by

2

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

= \frac{2}{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{2 - π}{6}

x > \frac{2 - π}{6} + \frac{2 π}{3} n

x > \frac{2 - π}{6} + \frac{2 π}{3} n

3 x - 1 < arccos (0) + 2 πn : x < \frac{2 + π}{6} + \frac{2 π}{3} n

3 x - 1 < 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 - 1 < \frac{π}{2} + 2 πn

Move

1

to the right side

3 x - 1 < \frac{π}{2} + 2 πn

Add

1

to both sides

3 x - 1 + 1 < \frac{π}{2} + 2 πn + 1

Simplify

3 x < \frac{π}{2} + 2 πn + 1

3 x < \frac{π}{2} + 2 πn + 1

Divide both sides by

3

3 x < \frac{π}{2} + 2 πn + 1

Divide both sides by

3

\frac{3 x}{3} < \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Simplify

\frac{3 x}{3} < \frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3} : \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} + \frac{1}{3}

Group like terms

= \frac{1}{3} + \frac{2 πn}{3} + \frac{\frac{π}{2}}{3}

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{1}{3} + \frac{2 πn}{3} + \frac{π}{6}

Group like terms

= \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

x < \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

x < \frac{1}{3} + \frac{π}{6} + \frac{2 πn}{3}

Simplify

\frac{1}{3} + \frac{π}{6} : \frac{2 + π}{6}

\frac{1}{3} + \frac{π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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{1}{3} :

multiply the denominator and numerator by

2

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

= \frac{2}{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{2 + π}{6}

x < \frac{2 + π}{6} + \frac{2 π}{3} n

x < \frac{2 + π}{6} + \frac{2 π}{3} n

Combine the intervals

x > \frac{2 - π}{6} + \frac{2 π}{3} n and x < \frac{2 + π}{6} + \frac{2 π}{3} n

Merge Overlapping Intervals

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

Combine the intervals

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

Merge Overlapping Intervals

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

Popular Examples

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