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Popular >

cot((3x)/2)+sqrt(3)<= 0

Solution

cot (\frac{3 x}{2}) + 3 \leq 0

Solution

\frac{5 π}{9} + \frac{2 π}{3} n \leq x < \frac{2 π}{3} + \frac{2 π}{3} n

Interval Notation

[\frac{5 π}{9} + \frac{2 π}{3} n, \frac{2 π}{3} + \frac{2 π}{3} n)

Decimal Notation

1.74532 \dots + \frac{2 π}{3} n \leq x < 2.09439 \dots + \frac{2 π}{3} n

Solution steps

cot (\frac{3 x}{2}) + 3 \leq 0

Move

3

to the right side

cot (\frac{3 x}{2}) + 3 \leq 0

Subtract

3

from both sides

cot (\frac{3 x}{2}) + 3 - 3 \leq 0 - 3

Simplify

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

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

cot (x) \leq a

then

arccot (a) + πn \leq x < π + πn

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

a \leq u < b

then

a \leq u and u < b

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

arccot (- 3) + πn \leq \frac{3 x}{2} : x \geq \frac{5 π}{9} + \frac{2 πn}{3}

arccot (- 3) + πn \leq \frac{3 x}{2}

Switch sides

\frac{3 x}{2} \geq arccot (- 3) + πn

Simplify

arccot (- 3) + πn : \frac{5 π}{6} + πn

arccot (- 3) + πn

Use the following trivial identity:

arccot (- 3) = \frac{5 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{5 π}{6} + πn

\frac{3 x}{2} \geq \frac{5 π}{6} + πn

Multiply both sides by

2

\frac{3 x}{2} \geq \frac{5 π}{6} + πn

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} \geq 2 \cdot \frac{5 π}{6} + 2 πn

Simplify

\frac{2 \cdot 3 x}{2} \geq 2 \cdot \frac{5 π}{6} + 2 πn

Simplify

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

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot \frac{5 π}{6} + 2 πn : \frac{5 π}{3} + 2 πn

2 \cdot \frac{5 π}{6} + 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

= \frac{5 π}{3} + 2 πn

3 x \geq \frac{5 π}{3} + 2 πn

3 x \geq \frac{5 π}{3} + 2 πn

3 x \geq \frac{5 π}{3} + 2 πn

Divide both sides by

3

3 x \geq \frac{5 π}{3} + 2 πn

Divide both sides by

3

\frac{3 x}{3} \geq \frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 x}{3} \geq \frac{\frac{5 π}{3}}{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{5 π}{3}}{3} + \frac{2 πn}{3} : \frac{5 π}{9} + \frac{2 πn}{3}

\frac{\frac{5 π}{3}}{3} + \frac{2 πn}{3}

\frac{\frac{5 π}{3}}{3} = \frac{5 π}{9}

\frac{\frac{5 π}{3}}{3}

Apply the fraction rule:

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

= \frac{5 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{5 π}{9}

= \frac{5 π}{9} + \frac{2 πn}{3}

x \geq \frac{5 π}{9} + \frac{2 πn}{3}

x \geq \frac{5 π}{9} + \frac{2 πn}{3}

x \geq \frac{5 π}{9} + \frac{2 πn}{3}

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

\frac{3 x}{2} < π + πn

Multiply both sides by

2

\frac{3 x}{2} < π + πn

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} < 2 π + 2 πn

Simplify

3 x < 2 π + 2 πn

3 x < 2 π + 2 πn

Divide both sides by

3

3 x < 2 π + 2 πn

Divide both sides by

3

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

Simplify

x < \frac{2 π}{3} + \frac{2 πn}{3}

x < \frac{2 π}{3} + \frac{2 πn}{3}

Combine the intervals

x \geq \frac{5 π}{9} + \frac{2 πn}{3} and x < \frac{2 π}{3} + \frac{2 πn}{3}

Merge Overlapping Intervals

\frac{5 π}{9} + \frac{2 π}{3} n \leq x < \frac{2 π}{3} + \frac{2 π}{3} n

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cos (x) \geq sin (x)

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