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Popular Trigonometry >

tan(x/2-pi/3)>= 0

Solution

tan (\frac{x}{2} - \frac{π}{3}) \geq 0

Solution

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

Interval Notation

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

Decimal

2.09439 \dots + 2 πn \leq x < 5.23598 \dots + 2 πn

Solution steps

tan (\frac{x}{2} - \frac{π}{3}) \geq 0

tan (x) \geq a

then

arctan (a) + πn \leq x < \frac{π}{2} + πn

arctan (0) + πn \leq (\frac{x}{2} - \frac{π}{3}) < \frac{π}{2} + πn

a \leq u < b

then

a \leq u and u < b

arctan (0) + πn \leq \frac{x}{2} - \frac{π}{3} and \frac{x}{2} - \frac{π}{3} < \frac{π}{2} + πn

arctan (0) + πn \leq \frac{x}{2} - \frac{π}{3} : x \geq 2 πn + \frac{2 π}{3}

arctan (0) + πn \leq \frac{x}{2} - \frac{π}{3}

Switch sides

\frac{x}{2} - \frac{π}{3} \geq arctan (0) + πn

Simplify

arctan (0) + πn : πn

arctan (0) + πn

Use the following trivial identity:

arctan (0) = 0

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= 0 + πn

0 + πn = πn

= πn

\frac{x}{2} - \frac{π}{3} \geq πn

Move

\frac{π}{3}

to the right side

\frac{x}{2} - \frac{π}{3} \geq πn

Add

\frac{π}{3}

to both sides

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

Simplify

\frac{x}{2} \geq πn + \frac{π}{3}

\frac{x}{2} \geq πn + \frac{π}{3}

Multiply both sides by

2

\frac{x}{2} \geq πn + \frac{π}{3}

Multiply both sides by

2

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

Simplify

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

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}{3} : 2 πn + \frac{2 π}{3}

2 πn + 2 \cdot \frac{π}{3}

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= 2 πn + \frac{2 π}{3}

x \geq 2 πn + \frac{2 π}{3}

x \geq 2 πn + \frac{2 π}{3}

x \geq 2 πn + \frac{2 π}{3}

\frac{x}{2} - \frac{π}{3} < \frac{π}{2} + πn : x < 2 πn + \frac{5 π}{3}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

\frac{x}{2} - \frac{π}{3} + \frac{π}{3} : \frac{x}{2}

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

Add similar elements:

- \frac{π}{3} + \frac{π}{3} < 0

= \frac{x}{2}

Simplify

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

\frac{π}{2} + πn + \frac{π}{3}

Group like terms

= πn + \frac{π}{2} + \frac{π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 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}{2} :

multiply the denominator and numerator by

3

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

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π 2}{6}

Add similar elements:

3 π + 2 π = 5 π

= πn + \frac{5 π}{6}

\frac{x}{2} < πn + \frac{5 π}{6}

\frac{x}{2} < πn + \frac{5 π}{6}

\frac{x}{2} < πn + \frac{5 π}{6}

Multiply both sides by

2

\frac{x}{2} < πn + \frac{5 π}{6}

Multiply both sides by

2

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

Simplify

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

2 πn + 2 \cdot \frac{5 π}{6}

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}

= 2 πn + \frac{5 π}{3}

x < 2 πn + \frac{5 π}{3}

x < 2 πn + \frac{5 π}{3}

x < 2 πn + \frac{5 π}{3}

Combine the intervals

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

Merge Overlapping Intervals

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

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