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sqrt(3)cot(2x+1/2)<1

Solution

3 cot (2 x + \frac{1}{2}) < 1

Solution

\frac{2 π - 3}{12} + \frac{π}{2} n < x < \frac{2 π - 1}{4} + \frac{π}{2} n

Interval Notation

(\frac{2 π - 3}{12} + \frac{π}{2} n, \frac{2 π - 1}{4} + \frac{π}{2} n)

Decimal Notation

0.27359 \dots + \frac{π}{2} n < x < 1.32079 \dots + \frac{π}{2} n

Solution steps

3 cot (2 x + \frac{1}{2}) < 1

Divide both sides by

3

3 cot (2 x + \frac{1}{2}) < 1

Divide both sides by

3

\frac{3 cot ( 2 x + \frac{1}{2} )}{3} < \frac{1}{3}

Simplify

\frac{3 cot ( 2 x + \frac{1}{2} )}{3} < \frac{1}{3}

Simplify

\frac{3 cot ( 2 x + \frac{1}{2} )}{3} : cot (2 x + \frac{1}{2})

\frac{3 cot ( 2 x + \frac{1}{2} )}{3}

Cancel the common factor:

3

= cot (2 x + \frac{1}{2})

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

cot (2 x + \frac{1}{2}) < \frac{3}{3}

cot (2 x + \frac{1}{2}) < \frac{3}{3}

cot (2 x + \frac{1}{2}) < \frac{3}{3}

cot (x) < a

then

arccot (a) + πn < x < π + πn

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

a < u < b

then

a < u and u < b

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

arccot (\frac{3}{3}) + πn < 2 x + \frac{1}{2} : x > \frac{2 π - 3}{12} + \frac{π}{2} n

arccot (\frac{3}{3}) + πn < 2 x + \frac{1}{2}

Switch sides

2 x + \frac{1}{2} > arccot (\frac{3}{3}) + πn

Simplify

arccot (\frac{3}{3}) + πn : \frac{π}{3} + πn

arccot (\frac{3}{3}) + πn

Use the following trivial identity:

arccot (\frac{3}{3}) = \frac{π}{3}

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

= \frac{π}{3} + πn

2 x + \frac{1}{2} > \frac{π}{3} + πn

Move

\frac{1}{2}

to the right side

2 x + \frac{1}{2} > \frac{π}{3} + πn

Subtract

\frac{1}{2}

from both sides

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

Simplify

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

Simplify

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

2 x + \frac{1}{2} - \frac{1}{2}

Add similar elements:

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

= 2 x

Simplify

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

\frac{π}{3} + πn - \frac{1}{2}

Group like terms

= πn + \frac{π}{3} - \frac{1}{2}

Could not simplify further

= πn + \frac{π}{3} - \frac{1}{2}

2 x > πn + \frac{π}{3} - \frac{1}{2}

2 x > πn + \frac{π}{3} - \frac{1}{2}

2 x > πn + \frac{π}{3} - \frac{1}{2}

Divide both sides by

2

2 x > πn + \frac{π}{3} - \frac{1}{2}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{\frac{1}{2}}{2} = \frac{1}{4}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{πn}{2} + \frac{π}{6} - \frac{1}{4}

x > \frac{πn}{2} + \frac{π}{6} - \frac{1}{4}

x > \frac{πn}{2} + \frac{π}{6} - \frac{1}{4}

Simplify

\frac{π}{6} - \frac{1}{4} : \frac{2 π - 3}{12}

\frac{π}{6} - \frac{1}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

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

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

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 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{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{1}{4} :

multiply the denominator and numerator by

3

\frac{1}{4} = \frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}

= \frac{π 2}{12} - \frac{3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - 3}{12}

x > \frac{2 π - 3}{12} + \frac{π}{2} n

x > \frac{2 π - 3}{12} + \frac{π}{2} n

2 x + \frac{1}{2} < π + πn : x < \frac{2 π - 1}{4} + \frac{π}{2} n

2 x + \frac{1}{2} < π + πn

Move

\frac{1}{2}

to the right side

2 x + \frac{1}{2} < π + πn

Subtract

\frac{1}{2}

from both sides

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

Simplify

2 x < π + πn - \frac{1}{2}

2 x < π + πn - \frac{1}{2}

Divide both sides by

2

2 x < π + πn - \frac{1}{2}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} + \frac{πn}{2} - \frac{\frac{1}{2}}{2} : \frac{π}{2} - \frac{1}{4} + \frac{πn}{2}

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

\frac{\frac{1}{2}}{2} = \frac{1}{4}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= \frac{π}{2} + \frac{πn}{2} - \frac{1}{4}

Group like terms

= \frac{π}{2} - \frac{1}{4} + \frac{πn}{2}

x < \frac{π}{2} - \frac{1}{4} + \frac{πn}{2}

x < \frac{π}{2} - \frac{1}{4} + \frac{πn}{2}

Simplify

\frac{π}{2} - \frac{1}{4} : \frac{2 π - 1}{4}

\frac{π}{2} - \frac{1}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

\frac{π}{2} = \frac{π 2}{2 \cdot 2} = \frac{π 2}{4}

= \frac{π 2}{4} - \frac{1}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - 1}{4}

x < \frac{2 π - 1}{4} + \frac{π}{2} n

x < \frac{2 π - 1}{4} + \frac{π}{2} n

Combine the intervals

x > \frac{2 π - 3}{12} + \frac{π}{2} n and x < \frac{2 π - 1}{4} + \frac{π}{2} n

Merge Overlapping Intervals

\frac{2 π - 3}{12} + \frac{π}{2} n < x < \frac{2 π - 1}{4} + \frac{π}{2} n

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