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Popular >

1-2cos(x)<0[0.2pi]

Solution

1 - 2 cos (x) < 0 [0.2 π]

Solution

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

+2

Interval Notation

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

Decimal Notation

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

Solution steps

1 - 2 cos (x) < 0 \cdot [0.2 π]

Remove parentheses:

(a) = a

1 - 2 cos (x) < 0 \cdot 0.2 π

Apply rule

0 \cdot a = 0

1 - 2 cos (x) < 0

Move

1

to the right side

1 - 2 cos (x) < 0

Subtract

1

from both sides

1 - 2 cos (x) - 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 \leq a < 1

then

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

- arccos (\frac{1}{2}) + 2 πn < x < 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

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}

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

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