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Popular >

sqrt(3)tan^2(x)+3tan(x)>0

Solution

3 tan^{2} (x) + 3 tan (x) > 0

Solution

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

Interval Notation

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

Decimal Notation

πn < x < 1.57079 \dots + πn or - 1.57079 \dots + πn < x < - 1.04719 \dots + πn

Solution steps

3 tan^{2} (x) + 3 tan (x) > 0

Let:

u = tan (x)

3 u^{2} + 3 u > 0

3 u^{2} + 3 u > 0 : u < - 3 or u > 0

3 u^{2} + 3 u > 0

Factor

3 u^{2} + 3 u : u (3 u + 3)

3 u^{2} + 3 u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= 3 uu + 3 u

Factor out common term

u

= u (1 \cdot 3 u + 3)

Multiply the numbers:

1 \cdot 3 = 3

= u (3 u + 3)

u (3 u + 3) > 0

Identify the intervals

Find the signs of the factors of

u (3 u + 3)

Find the signs of

u

u = 0

u < 0

u > 0

Find the signs of

3 u + 3

3 u + 3 = 0 : u = - 3

3 u + 3 = 0

Move

3

to the right side

3 u + 3 = 0

Subtract

3

from both sides

3 u + 3 - 3 = 0 - 3

Simplify

3 u = - 3

3 u = - 3

Divide both sides by

3

3 u = - 3

Divide both sides by

3

\frac{3 u}{3} = \frac{- 3}{3}

Simplify

\frac{3 u}{3} = \frac{- 3}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 3}{3} : - 3

\frac{- 3}{3}

Apply the fraction rule:

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

= - \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

u = - 3

u = - 3

u = - 3

3 u + 3 < 0 : u < - 3

3 u + 3 < 0

Move

3

to the right side

3 u + 3 < 0

Subtract

3

from both sides

3 u + 3 - 3 < 0 - 3

Simplify

3 u < - 3

3 u < - 3

Divide both sides by

3

3 u < - 3

Divide both sides by

3

\frac{3 u}{3} < \frac{- 3}{3}

Simplify

\frac{3 u}{3} < \frac{- 3}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 3}{3} : - 3

\frac{- 3}{3}

Apply the fraction rule:

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

= - \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

u < - 3

u < - 3

u < - 3

3 u + 3 > 0 : u > - 3

3 u + 3 > 0

Move

3

to the right side

3 u + 3 > 0

Subtract

3

from both sides

3 u + 3 - 3 > 0 - 3

Simplify

3 u > - 3

3 u > - 3

Divide both sides by

3

3 u > - 3

Divide both sides by

3

\frac{3 u}{3} > \frac{- 3}{3}

Simplify

\frac{3 u}{3} > \frac{- 3}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{- 3}{3} : - 3

\frac{- 3}{3}

Apply the fraction rule:

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

= - \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

u > - 3

u > - 3

u > - 3

Summarize in a table:

u 3 u + 3 u (3 u + 3) u < - 3 - - + u = - 3 - 00 - 3 < u < 0 - + - u = 0 0 + 0 u > 0 + + +

Identify the intervals that satisfy the required condition:

> 0

u < - 3 or u > 0

u < - 3 or u > 0

u < - 3 or u > 0

Substitute back

u = tan (x)

tan (x) < - 3 or tan (x) > 0

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

tan (x) < - 3

tan (x) < a

then

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

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

Simplify

arctan (- 3) : - \frac{π}{3}

arctan (- 3)

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 3) = - arctan (3)

= - arctan (3)

Use the following trivial identity:

arctan (3) = \frac{π}{3}

arctan (3)

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}{3}

= - \frac{π}{3}

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

tan (x) > 0 : πn < x < \frac{π}{2} + πn

tan (x) > 0

tan (x) > a

then

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

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

Simplify

arctan (0) : 0

arctan (0)

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

0 + πn < x < \frac{π}{2} + πn

Simplify

πn < x < \frac{π}{2} + πn

Combine the intervals

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

Merge Overlapping Intervals

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

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