English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

2cos^3(3x)-cos(3x)<0

Solution

2 cos^{3} (3 x) - cos (3 x) < 0

Solution

\frac{π}{12} + \frac{2 π}{3} n < x < \frac{π}{6} + \frac{2 π}{3} n or \frac{π}{4} + \frac{2 π}{3} n < x < \frac{5 π}{12} + \frac{2 π}{3} n or \frac{π}{2} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n

Interval Notation

(\frac{π}{12} + \frac{2 π}{3} n, \frac{π}{6} + \frac{2 π}{3} n) \cup (\frac{π}{4} + \frac{2 π}{3} n, \frac{5 π}{12} + \frac{2 π}{3} n) \cup (\frac{π}{2} + \frac{2 π}{3} n, \frac{7 π}{12} + \frac{2 π}{3} n)

Decimal

0.26179 \dots + \frac{2 π}{3} n < x < 0.52359 \dots + \frac{2 π}{3} n or 0.78539 \dots + \frac{2 π}{3} n < x < 1.30899 \dots + \frac{2 π}{3} n or 1.57079 \dots + \frac{2 π}{3} n < x < 1.83259 \dots + \frac{2 π}{3} n

Solution steps

2 cos^{3} (3 x) - cos (3 x) < 0

Let:

u = cos (3 x)

2 u^{3} - u < 0

2 u^{3} - u < 0 : u < - \frac{2}{2} or 0 < u < \frac{2}{2}

2 u^{3} - u < 0

Factor

2 u^{3} - u : u (2 u + 1) (2 u - 1)

2 u^{3} - u

Factor out common term

u : u (2 u^{2} - 1)

2 u^{3} - u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u^{2} u

= 2 u^{2} u - u

Factor out common term

u

= u (2 u^{2} - 1)

= u (2 u^{2} - 1)

Factor

2 u^{2} - 1 : (2 u + 1) (2 u - 1)

2 u^{2} - 1

Rewrite

2 u^{2} - 1

(2 u)^{2} - 1^{2}

2 u^{2} - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} u^{2} - 1

Rewrite

1

1^{2}

= (2)^{2} u^{2} - 1^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} u^{2} = (2 u)^{2}

= (2 u)^{2} - 1^{2}

= (2 u)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 u)^{2} - 1^{2} = (2 u + 1) (2 u - 1)

= (2 u + 1) (2 u - 1)

= u (2 u + 1) (2 u - 1)

u (2 u + 1) (2 u - 1) < 0

Identify the intervals

Find the signs of the factors of

u (2 u + 1) (2 u - 1)

Find the signs of

u

u = 0

u < 0

u > 0

Find the signs of

2 u + 1

2 u + 1 = 0 : u = - \frac{2}{2}

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

\frac{2 u}{2} = \frac{- 1}{2}

Simplify

\frac{2 u}{2} = \frac{- 1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

2 u + 1 < 0 : u < - \frac{2}{2}

2 u + 1 < 0

Move

1

to the right side

2 u + 1 < 0

Subtract

1

from both sides

2 u + 1 - 1 < 0 - 1

Simplify

2 u < - 1

2 u < - 1

Divide both sides by

2

2 u < - 1

Divide both sides by

2

\frac{2 u}{2} < \frac{- 1}{2}

Simplify

\frac{2 u}{2} < \frac{- 1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u < - \frac{2}{2}

u < - \frac{2}{2}

u < - \frac{2}{2}

2 u + 1 > 0 : u > - \frac{2}{2}

2 u + 1 > 0

Move

1

to the right side

2 u + 1 > 0

Subtract

1

from both sides

2 u + 1 - 1 > 0 - 1

Simplify

2 u > - 1

2 u > - 1

Divide both sides by

2

2 u > - 1

Divide both sides by

2

\frac{2 u}{2} > \frac{- 1}{2}

Simplify

\frac{2 u}{2} > \frac{- 1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u > - \frac{2}{2}

u > - \frac{2}{2}

u > - \frac{2}{2}

Find the signs of

2 u - 1

2 u - 1 = 0 : u = \frac{2}{2}

2 u - 1 = 0

Move

1

to the right side

2 u - 1 = 0

Add

1

to both sides

2 u - 1 + 1 = 0 + 1

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

\frac{2 u}{2} = \frac{1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

u = \frac{2}{2}

2 u - 1 < 0 : u < \frac{2}{2}

2 u - 1 < 0

Move

1

to the right side

2 u - 1 < 0

Add

1

to both sides

2 u - 1 + 1 < 0 + 1

Simplify

2 u < 1

2 u < 1

Divide both sides by

2

2 u < 1

Divide both sides by

2

\frac{2 u}{2} < \frac{1}{2}

Simplify

\frac{2 u}{2} < \frac{1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

u < \frac{2}{2}

u < \frac{2}{2}

u < \frac{2}{2}

2 u - 1 > 0 : u > \frac{2}{2}

2 u - 1 > 0

Move

1

to the right side

2 u - 1 > 0

Add

1

to both sides

2 u - 1 + 1 > 0 + 1

Simplify

2 u > 1

2 u > 1

Divide both sides by

2

2 u > 1

Divide both sides by

2

\frac{2 u}{2} > \frac{1}{2}

Simplify

\frac{2 u}{2} > \frac{1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

u > \frac{2}{2}

u > \frac{2}{2}

u > \frac{2}{2}

Summarize in a table:

u 2 u + 1 2 u - 1 u (2 u + 1) (2 u - 1) u < - \frac{2}{2} - - - - u = - \frac{2}{2} - 0 - 0 - \frac{2}{2} < u < 0 - + - + u = 0 0 + - 0 0 < u < \frac{2}{2} + + - - u = \frac{2}{2} + + 00 u > \frac{2}{2} + + + +

Identify the intervals that satisfy the required condition:

< 0

u < - \frac{2}{2} or 0 < u < \frac{2}{2}

u < - \frac{2}{2} or 0 < u < \frac{2}{2}

u < - \frac{2}{2} or 0 < u < \frac{2}{2}

Substitute back

u = cos (3 x)

cos (3 x) < - \frac{2}{2} or 0 < cos (3 x) < \frac{2}{2}

cos (3 x) < - \frac{2}{2} : \frac{π}{4} + \frac{2 π}{3} n < x < \frac{5 π}{12} + \frac{2 π}{3} n

cos (3 x) < - \frac{2}{2}

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

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

a < u < b

then

a < u and u < b

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

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

arccos (- \frac{2}{2}) + 2 πn < 3 x

Switch sides

3 x > arccos (- \frac{2}{2}) + 2 πn

Simplify

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

arccos (- \frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{2}{2}) = \frac{3 π}{4}

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{3 π}{4} + 2 πn

3 x > \frac{3 π}{4} + 2 πn

Divide both sides by

3

3 x > \frac{3 π}{4} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

= \frac{π}{4} + \frac{2 πn}{3}

x > \frac{π}{4} + \frac{2 πn}{3}

x > \frac{π}{4} + \frac{2 πn}{3}

x > \frac{π}{4} + \frac{2 πn}{3}

3 x < 2 π - arccos (- \frac{2}{2}) + 2 πn : x < \frac{5 π}{12} + \frac{2 π}{3} n

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

Simplify

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

2 π - arccos (- \frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{2}{2}) = \frac{3 π}{4}

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{3 π}{4} + 2 πn

3 x < 2 π - \frac{3 π}{4} + 2 πn

Divide both sides by

3

3 x < 2 π - \frac{3 π}{4} + 2 πn

Divide both sides by

3

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

Simplify

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

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

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

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

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

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

Simplify

\frac{2 π}{3} - \frac{π}{4} : \frac{5 π}{12}

\frac{2 π}{3} - \frac{π}{4}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

3

4

= 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{2 π}{3} :

multiply the denominator and numerator by

4

\frac{2 π}{3} = \frac{2 π 4}{3 \cdot 4} = \frac{8 π}{12}

For

\frac{π}{4} :

multiply the denominator and numerator by

3

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

= \frac{8 π}{12} - \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{8 π - π 3}{12}

Add similar elements:

8 π - 3 π = 5 π

= \frac{5 π}{12}

x < \frac{5 π}{12} + \frac{2 π}{3} n

x < \frac{5 π}{12} + \frac{2 π}{3} n

Combine the intervals

x > \frac{π}{4} + \frac{2 πn}{3} and x < \frac{5 π}{12} + \frac{2 π}{3} n

Merge Overlapping Intervals

\frac{π}{4} + \frac{2 π}{3} n < x < \frac{5 π}{12} + \frac{2 π}{3} n

0 < cos (3 x) < \frac{2}{2} : \frac{π}{12} + \frac{2 π}{3} n < x < \frac{π}{6} + \frac{2 π}{3} n or \frac{π}{2} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n

0 < cos (3 x) < \frac{2}{2}

a < u < b

then

a < u and u < b

0 < cos (3 x) and cos (3 x) < \frac{2}{2}

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

0 < cos (3 x)

Switch sides

cos (3 x) > 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 < arccos (0) + 2 πn

a < u < b

then

a < u and u < b

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

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

- arccos (0) + 2 πn < 3 x

Switch sides

3 x > - 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}{2} + 2 πn

Divide both sides by

3

3 x > - \frac{π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

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

x > - \frac{π}{6} + \frac{2 πn}{3}

x > - \frac{π}{6} + \frac{2 πn}{3}

x > - \frac{π}{6} + \frac{2 πn}{3}

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

3 x < 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}{2} + 2 πn

Divide both sides by

3

3 x < \frac{π}{2} + 2 πn

Divide both sides by

3

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

Simplify

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

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

x < \frac{π}{6} + \frac{2 πn}{3}

x < \frac{π}{6} + \frac{2 πn}{3}

x < \frac{π}{6} + \frac{2 πn}{3}

Combine the intervals

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

Merge Overlapping Intervals

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

cos (3 x) < \frac{2}{2} : \frac{π}{12} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n

cos (3 x) < \frac{2}{2}

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

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

a < u < b

then

a < u and u < b

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

arccos (\frac{2}{2}) + 2 πn < 3 x : x > \frac{π}{12} + \frac{2 πn}{3}

arccos (\frac{2}{2}) + 2 πn < 3 x

Switch sides

3 x > arccos (\frac{2}{2}) + 2 πn

Simplify

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

arccos (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

3 x > \frac{π}{4} + 2 πn

Divide both sides by

3

3 x > \frac{π}{4} + 2 πn

Divide both sides by

3

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

Simplify

\frac{3 x}{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{\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 > \frac{π}{12} + \frac{2 πn}{3}

x > \frac{π}{12} + \frac{2 πn}{3}

x > \frac{π}{12} + \frac{2 πn}{3}

3 x < 2 π - arccos (\frac{2}{2}) + 2 πn : x < \frac{7 π}{12} + \frac{2 π}{3} n

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

Simplify

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

2 π - arccos (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

3 x < 2 π - \frac{π}{4} + 2 πn

Divide both sides by

3

3 x < 2 π - \frac{π}{4} + 2 πn

Divide both sides by

3

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

Simplify

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

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

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

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

Simplify

\frac{2 π}{3} - \frac{π}{12} : \frac{7 π}{12}

\frac{2 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{2 π}{3} :

multiply the denominator and numerator by

4

\frac{2 π}{3} = \frac{2 π 4}{3 \cdot 4} = \frac{8 π}{12}

= \frac{8 π}{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{8 π - π}{12}

Add similar elements:

8 π - π = 7 π

= \frac{7 π}{12}

x < \frac{7 π}{12} + \frac{2 π}{3} n

x < \frac{7 π}{12} + \frac{2 π}{3} n

Combine the intervals

x > \frac{π}{12} + \frac{2 πn}{3} and x < \frac{7 π}{12} + \frac{2 π}{3} n

Merge Overlapping Intervals

\frac{π}{12} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n

Combine the intervals

- \frac{π}{6} + \frac{2 π}{3} n < x < \frac{π}{6} + \frac{2 π}{3} n and \frac{π}{12} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n

Merge Overlapping Intervals

\frac{π}{12} + \frac{2 π}{3} n < x < \frac{π}{6} + \frac{2 π}{3} n or \frac{π}{2} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n

Combine the intervals

\frac{π}{4} + \frac{2 π}{3} n < x < \frac{5 π}{12} + \frac{2 π}{3} n or (\frac{π}{12} + \frac{2 π}{3} n < x < \frac{π}{6} + \frac{2 π}{3} n or \frac{π}{2} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n)

Merge Overlapping Intervals

\frac{π}{12} + \frac{2 π}{3} n < x < \frac{π}{6} + \frac{2 π}{3} n or \frac{π}{4} + \frac{2 π}{3} n < x < \frac{5 π}{12} + \frac{2 π}{3} n or \frac{π}{2} + \frac{2 π}{3} n < x < \frac{7 π}{12} + \frac{2 π}{3} n

Popular Examples

0<= sin(pix)

2cos^2(x)+sin(x)>2

0.5<= sin(30t)

sin(x)-sqrt(3)cos(x)>sqrt(2)

cos(x)<1+sin(x)