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-2cos(x)+1>0

Solution

- 2 cos (x) + 1 > 0

\frac{π}{3} + 2 πn < x < \frac{5 π}{3} + 2 πn

Interval Notation

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

Decimal

1.04719 \dots + 2 πn < x < 5.23598 \dots + 2 πn

Solution steps

- 2 cos (x) + 1 > 0

Move

1

to the right side

- 2 cos (x) + 1 > 0

Subtract

1

from both sides

- 2 cos (x) + 1 - 1 > 0 - 1

Simplify

- 2 cos (x) > - 1

- 2 cos (x) > - 1

Multiply both sides by

- 1

- 2 cos (x) > - 1

Multiply both sides by -1 (reverse the inequality)

(- 2 cos (x)) (- 1) < (- 1) (- 1)

Simplify

2 cos (x) < 1

2 cos (x) < 1

Divide both sides by

2

2 cos (x) < 1

Divide both sides by

2

\frac{2 cos ( x )}{2} < \frac{1}{2}

Simplify

cos (x) < \frac{1}{2}

cos (x) < \frac{1}{2}

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

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

Simplify

arccos (\frac{1}{2}) : \frac{π}{3}

arccos (\frac{1}{2})

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{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{π}{3}

Simplify

2 π - arccos (\frac{1}{2}) : \frac{5 π}{3}

2 π - arccos (\frac{1}{2})

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{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}

= 2 π - \frac{π}{3}

Simplify

2 π - \frac{π}{3}

Convert element to fraction:

2 π = \frac{2 π 3}{3}

= \frac{2 π 3}{3} - \frac{π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 3 - π}{3}

2 π 3 - π = 5 π

2 π 3 - π

Multiply the numbers:

2 \cdot 3 = 6

= 6 π - π

Add similar elements:

6 π - π = 5 π

= 5 π

= \frac{5 π}{3}

= \frac{5 π}{3}

\frac{π}{3} + 2 πn < x < \frac{5 π}{3} + 2 πn