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Popular >

cot(x/2+pi/6)>= 1/(sqrt(3))

Solution

cot (\frac{x}{2} + \frac{π}{6}) \geq \frac{1}{3}

Solution

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

Interval Notation

(- \frac{π}{3} + 2 πn, \frac{π}{3} + 2 πn]

Decimal Notation

- 1.04719 \dots + 2 πn < x \leq 1.04719 \dots + 2 πn

Solution steps

cot (\frac{x}{2} + \frac{π}{6}) \geq \frac{1}{3}

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 (\frac{x}{2} + \frac{π}{6}) \geq \frac{3}{3}

cot (x) \geq a

then

πn < x \leq arccot (a) + πn

πn < (\frac{x}{2} + \frac{π}{6}) \leq arccot (\frac{3}{3}) + πn

a < u \leq b

then

a < u and u \leq b

πn < \frac{x}{2} + \frac{π}{6} and \frac{x}{2} + \frac{π}{6} \leq arccot (\frac{3}{3}) + πn

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

πn < \frac{x}{2} + \frac{π}{6}

Switch sides

\frac{x}{2} + \frac{π}{6} > πn

Move

\frac{π}{6}

to the right side

\frac{x}{2} + \frac{π}{6} > πn

Subtract

\frac{π}{6}

from both sides

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

Simplify

\frac{x}{2} > πn - \frac{π}{6}

\frac{x}{2} > πn - \frac{π}{6}

Multiply both sides by

2

\frac{x}{2} > πn - \frac{π}{6}

Multiply both sides by

2

\frac{2 x}{2} > 2 πn - 2 \cdot \frac{π}{6}

Simplify

\frac{2 x}{2} > 2 πn - 2 \cdot \frac{π}{6}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 πn - 2 \cdot \frac{π}{6}

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

= 2 πn - \frac{π}{3}

x > 2 πn - \frac{π}{3}

x > 2 πn - \frac{π}{3}

x > 2 πn - \frac{π}{3}

\frac{x}{2} + \frac{π}{6} \leq arccot (\frac{3}{3}) + πn : x \leq 2 πn + \frac{π}{3}

\frac{x}{2} + \frac{π}{6} \leq 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

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

Move

\frac{π}{6}

to the right side

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

Subtract

\frac{π}{6}

from both sides

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

Simplify

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

Simplify

\frac{x}{2} + \frac{π}{6} - \frac{π}{6} : \frac{x}{2}

\frac{x}{2} + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} \leq 0

= \frac{x}{2}

Simplify

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

\frac{π}{3} + πn - \frac{π}{6}

Group like terms

= πn + \frac{π}{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{π}{3} :

multiply the denominator and numerator by

2

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

= \frac{π 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{π 2 - π}{6}

Add similar elements:

2 π - π = π

= πn + \frac{π}{6}

\frac{x}{2} \leq πn + \frac{π}{6}

\frac{x}{2} \leq πn + \frac{π}{6}

\frac{x}{2} \leq πn + \frac{π}{6}

Multiply both sides by

2

\frac{x}{2} \leq πn + \frac{π}{6}

Multiply both sides by

2

\frac{2 x}{2} \leq 2 πn + 2 \cdot \frac{π}{6}

Simplify

\frac{2 x}{2} \leq 2 πn + 2 \cdot \frac{π}{6}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 πn + 2 \cdot \frac{π}{6}

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

= 2 πn + \frac{π}{3}

x \leq 2 πn + \frac{π}{3}

x \leq 2 πn + \frac{π}{3}

x \leq 2 πn + \frac{π}{3}

Combine the intervals

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

Merge Overlapping Intervals

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

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