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Popular >

tan(2x)<3

Solution

tan (2 x) < 3

Solution

- \frac{π}{4} + \frac{π}{2} n < x < \frac{arctan ( 3 )}{2} + \frac{π}{2} n

+2

Interval Notation

(- \frac{π}{4} + \frac{π}{2} n, \frac{arctan ( 3 )}{2} + \frac{π}{2} n)

Decimal Notation

- 0.78539 \dots + \frac{π}{2} n < x < 0.62452 \dots + \frac{π}{2} n

Solution steps

tan (2 x) < 3

If

tan (x) < a

then

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

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

If

a < u < b

then

a < u and u < b

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

- \frac{π}{2} + πn < 2 x : x > - \frac{π}{4} + \frac{πn}{2}

- \frac{π}{2} + πn < 2 x

Switch sides

2 x > - \frac{π}{2} + πn

Divide both sides by

2

2 x > - \frac{π}{2} + πn

Divide both sides by

2

\frac{2 x}{2} > - \frac{\frac{π}{2}}{2} + \frac{πn}{2}

Simplify

\frac{2 x}{2} > - \frac{\frac{π}{2}}{2} + \frac{πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{\frac{π}{2}}{2} + \frac{πn}{2} : - \frac{π}{4} + \frac{πn}{2}

- \frac{\frac{π}{2}}{2} + \frac{πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= - \frac{π}{4} + \frac{πn}{2}

x > - \frac{π}{4} + \frac{πn}{2}

x > - \frac{π}{4} + \frac{πn}{2}

x > - \frac{π}{4} + \frac{πn}{2}

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

2 x < arctan (3) + πn

Divide both sides by

2

2 x < arctan (3) + πn

Divide both sides by

2

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

Simplify

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

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

Combine the intervals

x > - \frac{π}{4} + \frac{πn}{2} and x < \frac{arctan ( 3 )}{2} + \frac{πn}{2}

Merge Overlapping Intervals

- \frac{π}{4} + \frac{π}{2} n < x < \frac{arctan ( 3 )}{2} + \frac{π}{2} n

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