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Popular Trigonometry >

-1<tan(x/2)<-1/5

Solution

- 1 < tan (\frac{x}{2}) < - \frac{1}{5}

Solution

\frac{3 π}{2} + 2 πn < x < - 2 arctan (\frac{1}{5}) + 2 π + 2 πn

Interval Notation

(\frac{3 π}{2} + 2 πn, - 2 arctan (\frac{1}{5}) + 2 π + 2 πn)

Decimal

4.71238 \dots + 2 πn < x < 5.88839 \dots + 2 πn

Solution steps

- 1 < tan (\frac{x}{2}) < - \frac{1}{5}

a < u < b

then

a < u and u < b

- 1 < tan (\frac{x}{2}) and tan (\frac{x}{2}) < - \frac{1}{5}

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

- 1 < tan (\frac{x}{2})

Switch sides

tan (\frac{x}{2}) > - 1

tan (x) > a

then

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

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

a < u < b

then

a < u and u < b

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

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

arctan (- 1) + πn < \frac{x}{2}

Switch sides

\frac{x}{2} > arctan (- 1) + πn

Simplify

arctan (- 1) + πn : - \frac{π}{4} + πn

arctan (- 1) + πn

arctan (- 1) = - \frac{π}{4}

arctan (- 1)

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1) = - arctan (1)

= - arctan (1)

Use the following trivial identity:

arctan (1) = \frac{π}{4}

arctan (1)

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}

= - \frac{π}{4}

= - \frac{π}{4} + πn

\frac{x}{2} > - \frac{π}{4} + πn

Multiply both sides by

2

\frac{x}{2} > - \frac{π}{4} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

- 2 \cdot \frac{π}{4} + 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= - \frac{π}{2} + 2 πn

x > - \frac{π}{2} + 2 πn

x > - \frac{π}{2} + 2 πn

x > - \frac{π}{2} + 2 πn

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

\frac{x}{2} < \frac{π}{2} + πn

Multiply both sides by

2

\frac{x}{2} < \frac{π}{2} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{2} + 2 πn : π + 2 πn

2 \cdot \frac{π}{2} + 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= π + 2 πn

x < π + 2 πn

x < π + 2 πn

x < π + 2 πn

Combine the intervals

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

Merge Overlapping Intervals

- \frac{π}{2} + 2 πn < x < π + 2 πn

tan (\frac{x}{2}) < - \frac{1}{5} : - π + 2 πn < x < - 2 arctan (\frac{1}{5}) + 2 πn

tan (\frac{x}{2}) < - \frac{1}{5}

tan (x) < a

then

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

- \frac{π}{2} + πn < \frac{x}{2} < arctan (- \frac{1}{5}) + πn

a < u < b

then

a < u and u < b

- \frac{π}{2} + πn < \frac{x}{2} and \frac{x}{2} < arctan (- \frac{1}{5}) + πn

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

- \frac{π}{2} + πn < \frac{x}{2}

Switch sides

\frac{x}{2} > - \frac{π}{2} + πn

Multiply both sides by

2

\frac{x}{2} > - \frac{π}{2} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

- 2 \cdot \frac{π}{2} + 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= - π + 2 πn

x > - π + 2 πn

x > - π + 2 πn

x > - π + 2 πn

\frac{x}{2} < arctan (- \frac{1}{5}) + πn : x < - 2 arctan (\frac{1}{5}) + 2 πn

\frac{x}{2} < arctan (- \frac{1}{5}) + πn

Simplify

arctan (- \frac{1}{5}) + πn : - arctan (\frac{1}{5}) + πn

arctan (- \frac{1}{5}) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{1}{5}) = - arctan (\frac{1}{5})

= - arctan (\frac{1}{5}) + πn

\frac{x}{2} < - arctan (\frac{1}{5}) + πn

Multiply both sides by

2

\frac{x}{2} < - arctan (\frac{1}{5}) + πn

Multiply both sides by

2

\frac{2 x}{2} < - 2 arctan (\frac{1}{5}) + 2 πn

Simplify

x < - 2 arctan (\frac{1}{5}) + 2 πn

x < - 2 arctan (\frac{1}{5}) + 2 πn

Combine the intervals

x > - π + 2 πn and x < - 2 arctan (\frac{1}{5}) + 2 πn

Merge Overlapping Intervals

- π + 2 πn < x < - 2 arctan (\frac{1}{5}) + 2 πn

Combine the intervals

- \frac{π}{2} + 2 πn < x < π + 2 πn and - π + 2 πn < x < - 2 arctan (\frac{1}{5}) + 2 πn

Merge Overlapping Intervals

\frac{3 π}{2} + 2 πn < x < - 2 arctan (\frac{1}{5}) + 2 π + 2 πn

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