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Popular >

cos(x/2)>-1/2

Solution

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

Solution

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

Interval Notation

(- \frac{4 π}{3} + 4 πn, \frac{4 π}{3} + 4 πn)

Decimal Notation

- 4.18879 \dots + 4 πn < x < 4.18879 \dots + 4 πn

Solution steps

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

For

cos (x) > a

, if

- 1 \leq a < 1

then

- arccos (a) + 2 πn < x < arccos (a) + 2 πn

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

a < u < b

then

a < u and u < b

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

- arccos (- \frac{1}{2}) + 2 πn < \frac{x}{2} : x > - \frac{4 π}{3} + 4 πn

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

Switch sides

\frac{x}{2} > - arccos (- \frac{1}{2}) + 2 πn

Simplify

- arccos (- \frac{1}{2}) + 2 πn : - \frac{2 π}{3} + 2 πn

- arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= - \frac{2 π}{3} + 2 πn

\frac{x}{2} > - \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} > - \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} > - 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} > - 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn : - \frac{4 π}{3} + 4 πn

- 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= - \frac{4 π}{3} + 4 πn

x > - \frac{4 π}{3} + 4 πn

x > - \frac{4 π}{3} + 4 πn

x > - \frac{4 π}{3} + 4 πn

\frac{x}{2} < arccos (- \frac{1}{2}) + 2 πn : x < \frac{4 π}{3} + 4 πn

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

Simplify

arccos (- \frac{1}{2}) + 2 πn : \frac{2 π}{3} + 2 πn

arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{2 π}{3} + 2 πn

\frac{x}{2} < \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} < \frac{2 π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn : \frac{4 π}{3} + 4 πn

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{4 π}{3} + 4 πn

x < \frac{4 π}{3} + 4 πn

x < \frac{4 π}{3} + 4 πn

x < \frac{4 π}{3} + 4 πn

Combine the intervals

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

Merge Overlapping Intervals

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

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