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Popular >

sqrt(3)cot(x/2)>3

Solution

3 cot (\frac{x}{2}) > 3

Solution

2 πn < x < \frac{π}{3} + 2 πn

+2

Interval Notation

(2 πn, \frac{π}{3} + 2 πn)

Decimal Notation

2 πn < x < 1.04719 \dots + 2 πn

Solution steps

3 cot (\frac{x}{2}) > 3

Divide both sides by

3

3 cot (\frac{x}{2}) > 3

Divide both sides by

3

\frac{3 cot ( \frac{x}{2} )}{3} > \frac{3}{3}

Simplify

\frac{3 cot ( \frac{x}{2} )}{3} > \frac{3}{3}

Simplify

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

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

Cancel the common factor:

3

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

Simplify

\frac{3}{3} : 3

\frac{3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

= \frac{3}{3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtract the numbers:

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

= 3^{\frac{1}{2}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= 3

cot (\frac{x}{2}) > 3

cot (\frac{x}{2}) > 3

cot (\frac{x}{2}) > 3

If

cot (x) > a

then

πn < x < arccot (a) + πn

πn < \frac{x}{2} < arccot (3) + πn

If

a < u < b

then

a < u and u < b

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

πn < \frac{x}{2} : x > 2 πn

πn < \frac{x}{2}

Switch sides

\frac{x}{2} > πn

Multiply both sides by

2

\frac{x}{2} > πn

Multiply both sides by

2

\frac{2 x}{2} > 2 πn

Simplify

x > 2 πn

x > 2 πn

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

\frac{x}{2} < arccot (3) + πn

Simplify

arccot (3) + πn : \frac{π}{6} + πn

arccot (3) + πn

Use the following trivial identity:

arccot (3) = \frac{π}{6}

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}{6} + πn

\frac{x}{2} < \frac{π}{6} + πn

Multiply both sides by

2

\frac{x}{2} < \frac{π}{6} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{π}{6} + 2 πn

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}

= \frac{π}{3} + 2 πn

x < \frac{π}{3} + 2 πn

x < \frac{π}{3} + 2 πn

x < \frac{π}{3} + 2 πn

Combine the intervals

x > 2 πn and x < \frac{π}{3} + 2 πn

Merge Overlapping Intervals

2 πn < x < \frac{π}{3} + 2 πn

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