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Popular Trigonometry >

3cos(3 x/2-pi/4)-1>0

Solution

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

Solution

\frac{π - 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n < x < \frac{π + 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

Interval Notation

(\frac{π - 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n, \frac{π + 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n)

Decimal

- 0.29704 \dots + \frac{4 π}{3} n < x < 1.34423 \dots + \frac{4 π}{3} n

Solution steps

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

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

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

- arccos (\frac{1}{3}) + 2 πn < (3 \cdot \frac{x}{2} - \frac{π}{4}) < arccos (\frac{1}{3}) + 2 πn

a < u < b

then

a < u and u < b

- arccos (\frac{1}{3}) + 2 πn < 3 \cdot \frac{x}{2} - \frac{π}{4} and 3 \cdot \frac{x}{2} - \frac{π}{4} < arccos (\frac{1}{3}) + 2 πn

- arccos (\frac{1}{3}) + 2 πn < 3 \cdot \frac{x}{2} - \frac{π}{4} : x > \frac{π - 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

- arccos (\frac{1}{3}) + 2 πn < 3 \cdot \frac{x}{2} - \frac{π}{4}

Switch sides

3 \cdot \frac{x}{2} - \frac{π}{4} > - arccos (\frac{1}{3}) + 2 πn

Move

\frac{π}{4}

to the right side

3 \cdot \frac{x}{2} - \frac{π}{4} > - arccos (\frac{1}{3}) + 2 πn

Add

\frac{π}{4}

to both sides

3 \cdot \frac{x}{2} - \frac{π}{4} + \frac{π}{4} > - arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Simplify

3 \cdot \frac{x}{2} > - arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

3 \cdot \frac{x}{2} > - arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Refine

3 \cdot \frac{x}{2} : \frac{3 x}{2}

3 \cdot \frac{x}{2}

Multiply fractions:

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

= \frac{x \cdot 3}{2}

\frac{3 x}{2} > - arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{3 x}{2} > - arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{3 x}{2} \cdot 2 > - arccos (\frac{1}{3}) \cdot 2 + 2 πn \cdot 2 + \frac{π}{4} \cdot 2

Simplify

\frac{3 x}{2} \cdot 2 > - arccos (\frac{1}{3}) \cdot 2 + 2 πn \cdot 2 + \frac{π}{4} \cdot 2

Simplify

\frac{3 x}{2} \cdot 2 : 3 x

\frac{3 x}{2} \cdot 2

Multiply fractions:

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

= \frac{3 x \cdot 2}{2}

Cancel the common factor:

2

= 3 x

Simplify

arccos (\frac{1}{3}) \cdot 2 : 2 arccos (\frac{1}{3})

arccos (\frac{1}{3}) \cdot 2

Apply the commutative law:

arccos (\frac{1}{3}) \cdot 2 = 2 arccos (\frac{1}{3})

2 arccos (\frac{1}{3})

Simplify

2 πn \cdot 2 : 4 πn

2 πn \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

Simplify

\frac{π}{4} \cdot 2 : \frac{π}{2}

\frac{π}{4} \cdot 2

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

3 x > - 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

3 x > - 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

3 x > - 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

Divide both sides by

3

3 x > - 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

Divide both sides by

3

\frac{3 x}{3} > - \frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3}

Simplify

\frac{3 x}{3} > - \frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

- \frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3} : \frac{4 πn}{3} + \frac{π}{6} - \frac{2 arccos ( \frac{1}{3} )}{3}

- \frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3}

Group like terms

= \frac{4 πn}{3} + \frac{\frac{π}{2}}{3} - \frac{2 arccos ( \frac{1}{3} )}{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{4 πn}{3} + \frac{π}{6} - \frac{2 arccos ( \frac{1}{3} )}{3}

x > \frac{4 πn}{3} + \frac{π}{6} - \frac{2 arccos ( \frac{1}{3} )}{3}

x > \frac{4 πn}{3} + \frac{π}{6} - \frac{2 arccos ( \frac{1}{3} )}{3}

Simplify

\frac{π}{6} - \frac{2 arccos ( \frac{1}{3} )}{3} : \frac{π - 4 arccos ( \frac{1}{3} )}{6}

\frac{π}{6} - \frac{2 arccos ( \frac{1}{3} )}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

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

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 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{2 arccos ( \frac{1}{3} )}{3} :

multiply the denominator and numerator by

2

\frac{2 arccos ( \frac{1}{3} )}{3} = \frac{2 arccos ( \frac{1}{3} ) \cdot 2}{3 \cdot 2} = \frac{4 arccos ( \frac{1}{3} )}{6}

= \frac{π}{6} - \frac{4 arccos ( \frac{1}{3} )}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π - 4 arccos ( \frac{1}{3} )}{6}

x > \frac{π - 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

x > \frac{π - 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

3 \cdot \frac{x}{2} - \frac{π}{4} < arccos (\frac{1}{3}) + 2 πn : x < \frac{π + 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

3 \cdot \frac{x}{2} - \frac{π}{4} < arccos (\frac{1}{3}) + 2 πn

Move

\frac{π}{4}

to the right side

3 \cdot \frac{x}{2} - \frac{π}{4} < arccos (\frac{1}{3}) + 2 πn

Add

\frac{π}{4}

to both sides

3 \cdot \frac{x}{2} - \frac{π}{4} + \frac{π}{4} < arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Simplify

3 \cdot \frac{x}{2} < arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

3 \cdot \frac{x}{2} < arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Refine

3 \cdot \frac{x}{2} : \frac{3 x}{2}

3 \cdot \frac{x}{2}

Multiply fractions:

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

= \frac{x \cdot 3}{2}

\frac{3 x}{2} < arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{3 x}{2} < arccos (\frac{1}{3}) + 2 πn + \frac{π}{4}

Multiply both sides by

2

\frac{3 x}{2} \cdot 2 < arccos (\frac{1}{3}) \cdot 2 + 2 πn \cdot 2 + \frac{π}{4} \cdot 2

Simplify

\frac{3 x}{2} \cdot 2 < arccos (\frac{1}{3}) \cdot 2 + 2 πn \cdot 2 + \frac{π}{4} \cdot 2

Simplify

\frac{3 x}{2} \cdot 2 : 3 x

\frac{3 x}{2} \cdot 2

Multiply fractions:

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

= \frac{3 x \cdot 2}{2}

Cancel the common factor:

2

= 3 x

Simplify

arccos (\frac{1}{3}) \cdot 2 : 2 arccos (\frac{1}{3})

arccos (\frac{1}{3}) \cdot 2

Apply the commutative law:

arccos (\frac{1}{3}) \cdot 2 = 2 arccos (\frac{1}{3})

2 arccos (\frac{1}{3})

Simplify

2 πn \cdot 2 : 4 πn

2 πn \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

Simplify

\frac{π}{4} \cdot 2 : \frac{π}{2}

\frac{π}{4} \cdot 2

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

3 x < 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

3 x < 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

3 x < 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

Divide both sides by

3

3 x < 2 arccos (\frac{1}{3}) + 4 πn + \frac{π}{2}

Divide both sides by

3

\frac{3 x}{3} < \frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3}

Simplify

\frac{3 x}{3} < \frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3} : \frac{4 πn}{3} + \frac{π}{6} + \frac{2 arccos ( \frac{1}{3} )}{3}

\frac{2 arccos ( \frac{1}{3} )}{3} + \frac{4 πn}{3} + \frac{\frac{π}{2}}{3}

Group like terms

= \frac{4 πn}{3} + \frac{\frac{π}{2}}{3} + \frac{2 arccos ( \frac{1}{3} )}{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{4 πn}{3} + \frac{π}{6} + \frac{2 arccos ( \frac{1}{3} )}{3}

x < \frac{4 πn}{3} + \frac{π}{6} + \frac{2 arccos ( \frac{1}{3} )}{3}

x < \frac{4 πn}{3} + \frac{π}{6} + \frac{2 arccos ( \frac{1}{3} )}{3}

Simplify

\frac{π}{6} + \frac{2 arccos ( \frac{1}{3} )}{3} : \frac{π + 4 arccos ( \frac{1}{3} )}{6}

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

Least Common Multiplier of

6, 3 : 6

6, 3

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

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 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{2 arccos ( \frac{1}{3} )}{3} :

multiply the denominator and numerator by

2

\frac{2 arccos ( \frac{1}{3} )}{3} = \frac{2 arccos ( \frac{1}{3} ) \cdot 2}{3 \cdot 2} = \frac{4 arccos ( \frac{1}{3} )}{6}

= \frac{π}{6} + \frac{4 arccos ( \frac{1}{3} )}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 4 arccos ( \frac{1}{3} )}{6}

x < \frac{π + 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

x < \frac{π + 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

Combine the intervals

x > \frac{π - 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n and x < \frac{π + 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

Merge Overlapping Intervals

\frac{π - 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n < x < \frac{π + 4 arccos ( \frac{1}{3} )}{6} + \frac{4 π}{3} n

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