English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

tan(x+pi/3)>1

Solution

tan (x + \frac{π}{3}) > 1

Solution

- \frac{π}{12} + πn < x < \frac{π}{6} + πn

Interval Notation

(- \frac{π}{12} + πn, \frac{π}{6} + πn)

Decimal

- 0.26179 \dots + πn < x < 0.52359 \dots + πn

Solution steps

tan (x + \frac{π}{3}) > 1

tan (x) > a

then

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

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

a < u < b

then

a < u and u < b

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

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

arctan (1) + πn < x + \frac{π}{3}

Switch sides

x + \frac{π}{3} > arctan (1) + πn

Simplify

arctan (1) + πn : \frac{π}{4} + πn

arctan (1) + πn

Use the following trivial identity:

arctan (1) = \frac{π}{4}

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}

= \frac{π}{4} + πn

x + \frac{π}{3} > \frac{π}{4} + πn

Move

\frac{π}{3}

to the right side

x + \frac{π}{3} > \frac{π}{4} + πn

Subtract

\frac{π}{3}

from both sides

x + \frac{π}{3} - \frac{π}{3} > \frac{π}{4} + πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} > \frac{π}{4} + πn - \frac{π}{3}

Simplify

x + \frac{π}{3} - \frac{π}{3} : x

x + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} > 0

= x

Simplify

\frac{π}{4} + πn - \frac{π}{3} : πn - \frac{π}{12}

\frac{π}{4} + πn - \frac{π}{3}

Group like terms

= πn + \frac{π}{4} - \frac{π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 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

4

3

= 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}{4} :

multiply the denominator and numerator by

3

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

For

\frac{π}{3} :

multiply the denominator and numerator by

4

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

= \frac{π 3}{12} - \frac{π 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π 4}{12}

Add similar elements:

3 π - 4 π = - π

= \frac{- π}{12}

Apply the fraction rule:

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

= πn - \frac{π}{12}

x > πn - \frac{π}{12}

x > πn - \frac{π}{12}

x > πn - \frac{π}{12}

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

x + \frac{π}{3} < \frac{π}{2} + πn

Move

\frac{π}{3}

to the right side

x + \frac{π}{3} < \frac{π}{2} + πn

Subtract

\frac{π}{3}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{3} - \frac{π}{3} : x

x + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} < 0

= x

Simplify

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

= πn + \frac{π}{6}

x < πn + \frac{π}{6}

x < πn + \frac{π}{6}

x < πn + \frac{π}{6}

Combine the intervals

x > πn - \frac{π}{12} and x < πn + \frac{π}{6}

Merge Overlapping Intervals

- \frac{π}{12} + πn < x < \frac{π}{6} + πn

Popular Examples

2cos^2(x)-3cos(x)+1<0