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Popular Trigonometry >

3tan^2(x)>1

Solution

3 tan^{2} (x) > 1

Solution

\frac{π}{6} + πn < x < \frac{π}{2} + πn or - \frac{π}{2} + πn < x < - \frac{π}{6} + πn

+2

Interval Notation

(\frac{π}{6} + πn, \frac{π}{2} + πn) \cup (- \frac{π}{2} + πn, - \frac{π}{6} + πn)

Decimal

0.52359 \dots + πn < x < 1.57079 \dots + πn or - 1.57079 \dots + πn < x < - 0.52359 \dots + πn

Solution steps

3 tan^{2} (x) > 1

Divide both sides by

3

3 tan^{2} (x) > 1

Divide both sides by

3

\frac{3 tan ^{2} ( x )}{3} > \frac{1}{3}

Simplify

tan^{2} (x) > \frac{1}{3}

tan^{2} (x) > \frac{1}{3}

For

u^{n} > a

, if

n

is even then

tan (x) < - \frac{1}{3} or tan (x) > \frac{1}{3}

tan (x) < - \frac{1}{3} : - \frac{π}{2} + πn < x < - \frac{π}{6} + πn

tan (x) < - \frac{1}{3}

If

tan (x) < a

then

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

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

Simplify

arctan (- \frac{1}{3}) : - \frac{π}{6}

arctan (- \frac{1}{3})

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{1}{3}) = - arctan (\frac{1}{3})

= - arctan (\frac{1}{3})

Use the following trivial identity:

arctan (\frac{1}{3}) = \frac{π}{6}

arctan (\frac{1}{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}{6}

= - \frac{π}{6}

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

tan (x) > \frac{1}{3} : \frac{π}{6} + πn < x < \frac{π}{2} + πn

tan (x) > \frac{1}{3}

If

tan (x) > a

then

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

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

Simplify

arctan (\frac{1}{3}) : \frac{π}{6}

arctan (\frac{1}{3})

Use the following trivial identity:

arctan (\frac{1}{3}) = \frac{π}{6}

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}{6}

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

Combine the intervals

- \frac{π}{2} + πn < x < - \frac{π}{6} + πn or \frac{π}{6} + πn < x < \frac{π}{2} + πn

Merge Overlapping Intervals

\frac{π}{6} + πn < x < \frac{π}{2} + πn or - \frac{π}{2} + πn < x < - \frac{π}{6} + πn

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